P. Colli, A. Favini, E. Rocca, G. Schimperna, J. Sprekels, eds., Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs: In Honour of Prof. Gianni Gilardi, 22 of Springer INdAM Series, Springer International Publishing AG, Cham, 2017, xii+571 pages, (Collection Published).AbstractThis volume gathers contributions in the field of partial differential equations, with a focus on mathematical models in phase transitions, complex fluids and thermomechanics. These contributions are dedicated to Professor Gianni Gilardi on the occasion of his 70th birthday. It particularly develops the following thematic areas: nonlinear dynamic and stationary equations; well-posedness of initial and boundary value problems for systems of PDEs; regularity properties for the solutions; optimal control problems and optimality conditions; feedback stabilization and stability results. Most of the articles are presented in a self-contained manner, and describe new achievements and/or the state of the art in their line of research, providing interested readers with an overview of recent advances and future research directions in PDEs.

C. Bucur, E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, Springer International Publishing Switzerland, Cham, 2016, xii+155 pages, (Monograph Published).AbstractWorking in the fractional Laplace framework, this book provides models and theorems related to nonlocal diffusion phenomena. In addition to a simple probabilistic interpretation, some applications to water waves, crystal dislocations, nonlocal phase transitions, nonlocal minimal surfaces and Schrödinger equations are given. Furthermore, an example of an s-harmonic function, its harmonic extension and some insight into a fractional version of a classical conjecture due to De Giorgi are presented. Although the aim is primarily to gather some introductory material concerning applications of the fractional Laplacian, some of the proofs and results are new. The work is entirely self-contained, and readers who wish to pursue related subjects of interest are invited to consult the rich bibliography for guidance.

A. Mielke, Chapter 3: On Evolutionary $Gamma$-Convergence for Gradient Systems, in: Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity, A. Muntean, J.D.M. Rademacher, A. Zagaris, eds., 3 of Lecture Notes in Applied Mathematics and Mechanics, Springer International Publishing Switzerland, Cham, 2016, pp. 187--249, (Chapter Published).AbstractIn these notes we discuss general approaches for rigorously deriving limits of generalized gradient flows. Our point of view is that a generalized gradient system is defined in terms of two functionals, namely the energy functional Eε and the dissipation potential Rε or the associated dissipation distance. We assume that the functionals depend on a small parameter and the associated gradients systems have solutions uε. We investigate the question under which conditions the limits u of (subsequences of) the solutions uε are solutions of the gradient system generated by the Γ-limits E0 and R0. Here the choice of the right topology will be crucial as well as additional structural conditions. We cover classical gradient systems, where Rε is quadratic, and rate-independent systems as well as the passage from viscous to rate-independent systems. Various examples, such as periodic homogenization, are used to illustrate the abstract concepts and results.

A. Mielke, T. Roubíček, Rate-independent Systems. Theory and Application, 193 of Applied Mathematical Sciences, Springer International Publishing, New York, 2015, vii+660 pages, (Monograph Published).AbstractThis monograph provides both an introduction to and a thorough exposition of the theory of rate-independent systems, which the authors have been working on with a lot of collaborators over 15 years. The focus is mostly on fully rate-independent systems, first on an abstract level either with or even without a linear structure, discussing various concepts of solutions with full mathematical rigor. Then, usefulness of the abstract concepts is demonstrated on the level of various applications primarily in continuum mechanics of solids, including suitable approximation strategies with guaranteed numerical stability and convergence. Particular applications concern inelastic processes such as plasticity, damage, phase transformations, or adhesive-type contacts both at small strains and at finite strains. A few other physical systems, e.g. magnetic or ferroelectric materials, and couplings to rate-dependent thermodynamic models are considered as well. Selected applications are accompanied by numerical simulations illustrating both the models and the efficiency of computational algorithms. In this book, the mathematical framework for a rigorous mathematical treatment of "rate-independent systems" is presented in a comprehensive form for the first time. Researchers and graduate students in applied mathematics, engineering, and computational physics will find this timely and well written book useful.

A. Mielke, Chapter 5: Variational Approaches and Methods for Dissipative Material Models with Multiple Scales, in: Analysis and Computation of Microstructure in Finite Plasticity, S. Conti, K. Hackl, eds., 78 of Lecture Notes in Applied and Computational Mechanics, Springer International Publishing, Heidelberg et al., 2015, pp. 125--155, (Chapter Published).AbstractIn a first part we consider evolutionary systems given as generalized gradient systems and discuss various variational principles that can be used to construct solutions for a given system or to derive the limit dynamics for multiscale problems. These multiscale limits are formulated in the theory of evolutionary Gamma-convergence. On the one hand we consider the a family of viscous gradient system with quadratic dissipation potentials and a wiggly energy landscape that converge to a rate-independent system. On the other hand we show how the concept of Balanced-Viscosity solution arise as in the vanishing-viscosity limit.

As applications we discuss, first, the evolution of laminate microstructures in finite-strain elastoplasticity and, second, a two-phase model for shape-memory materials, where H-measures are used to construct the mutual recovery sequences needed in the existence theory.

A. Mielke, Chapter 21: Dissipative Quantum Mechanics Using GENERIC, in: Recent Trends in Dynamical Systems -- Proceedings of a Conference in Honor of Jürgen Scheurle, A. Johann, H.-P. Kruse, F. Rupp, S. Schmitz, eds., 35 of Springer Proceedings in Mathematics & Statistics, Springer, Basel et al., 2013, pp. 555--585, (Chapter Published).AbstractPure quantum mechanics can be formulated as a Hamiltonian system in terms of the density matrix. Dissipative effects are modeled via coupling to a macroscopic system, where the coupling operators act via commutators. Following Öttinger (2010) we use the GENERIC framework (General Equations for Non-Equilibrium Reversible Irreversible Coupling) to construct thermodynamically consistent evolution equations as a sum of a Hamiltonian and a gradient-flow contribution, which satisfy a particular non-interaction condition. One of our models couples a quantum system to a finite number of heat baths each of which is described by a time-dependent temperature. The dissipation mechanism is modeled via the canonical correlation operator, which is the inverse of the Kubo-Mori metric for density matrices and which is strongly linked to the von Neumann entropy for quantum systems. Thus, one recovers the dissipative double-bracket operators of the Lindblad equations but encounters a correction term for the consistent coupling to the dissipative dynamics. For the finite-dimensional and isothermal case we provide a general existence result and discuss sufficient conditions that guarantee that all solutions converge to the unique thermal equilibrium state. Finally, we compare of our gradient flow formulation for quantum systems with the Wasserstein gradient flow formulation for the Fokker-Planck equation and the entropy gradient flow formulation for reversible Markov chains.

D. Belomestny, J.G.M. Schoenmakers, Optimal stopping of McKean-Vlasov diffusions via regression on particle systems, SIAM Journal on Control and Optimization, 58 (2020), pp. 529--550, DOI 10.1137/18M1195590 .AbstractIn this note we consider the problem of using regression on interacting particles to compute conditional expectations for McKean-Vlasov SDEs. We prove general result on convergence of linear regression algorithms and establish the corresponding rates of convergence. Application to optimal stopping and variance reduction are considered.

H. Meinlschmidt, Ch. Meyer, J. Rehberg, Regularization for optimal control problems associated to nonlinear evolution equations, Journal of Convex Analysis, 27 (2020), pp. 443--485, DOI 10.20347/WIAS.PREPRINT.2576 .AbstractIt is well-known that in the case of a sufficiently nonlinear general optimal control problem there is very frequently the necessity for a compactness argument in order to pass to the limit in the state equation in the standard “calculus of variations” proof for the existence of optimal controls. For time-dependent state equations, i.e., evolution equations, this is in particular unfortunate due to the difficult structure of compact sets in Bochner-type spaces. In this paper, we propose an abstract function space and a suitable regularization- or Tychonov term for the objective functional which allows for the usual standard reasoning in the proof of existence of optimal controls and which admits a reasonably favorable structure in the characterization of optimal solutions via first order necessary conditions in, generally, the form of a variational inequality of obstacle-type in time. We establish the necessary properties of the function space and the Tychonov term and derive the aforementioned variational inequality. The variational inequality can then be reformulated as a projection identity for the optimal control under additional assumptions. We give sufficient conditions on when these are satisfied. The considerations are complemented with a series of practical examples of possible constellations and choices in dependence on the varying control spaces required for the evolution equations at hand.

H. Neidhardt, A. Stephan, V.A. Zagrebnov, Convergence rate estimates for Trotter product approximations of solution operators for non-autonomous Cauchy problems, Publications of the Research Institute for Mathematical Sciences, 56 (2020), pp. 83--135 (published online 21.01.2020), DOI 10.4171/PRIMS/56-1-5 .AbstractIn the present paper we advocate the Howland-Evans approach to solution of the abstract non-autonomous Cauchy problem (non-ACP) in a separable Banach space X. The main idea is to reformulate this problem as an autonomous Cauchy problem (ACP) in a new Banach space Lp(J,X), consisting of X-valued functions on the time-interval J. The fundamental observation is a one-to-one correspondence between solution operators (propagators) for a non-ACP and the corresponding evolution semigroups for ACP in Lp(J,X). We show that the latter also allows to apply a full power of the operator-theoretical methods to scrutinise the non-ACP including the proof of the Trotter product approximation formulae with operator-norm estimate of the rate of convergence. The paper extends and improves some recent results in this direction in particular for Hilbert spaces.

A. Alphonse, M. Hintermüller, C.N. Rautenberg, Existence, iteration procedures and directional differentiability for parabolic QVIs, Calculus of Variations and Partial Differential Equations, 59 (2020), pp. 95/1--95/53, DOI 10.1007/s00526-020-01732-6 .AbstractWe study parabolic quasi-variational inequalities (QVIs) of obstacle type. Under appropriate assumptions on the obstacle mapping, we prove the existence of solutions of such QVIs by two methods: one by time discretisation through elliptic QVIs and the second by iteration through parabolic variational inequalities (VIs). Using these results, we show the directional differentiability (in a certain sense) of the solution map which takes the source term of a parabolic QVI into the set of solutions, and we relate this result to the contingent derivative of the aforementioned map. We finish with an example where the obstacle mapping is given by the inverse of a parabolic differential operator.

K. Disser, J. Rehberg, The 3D transient semiconductor equations with gradient-dependent and interfacial recombination, Mathematical Models & Methods in Applied Sciences, 29 (2019), pp. 1819--1851, DOI 10.1142/S0218202519500350 .AbstractWe establish the well-posedness of the transient van Roosbroeck system in three space dimensions under realistic assumptions on the data: non-smooth domains, discontinuous coefficient functions and mixed boundary conditions. Moreover, within this analysis, recombination terms may be concentrated on surfaces and interfaces and may not only depend on charge-carrier densities, but also on the electric field and currents. In particular, this includes Avalanche recombination. The proofs are based on recent abstract results on maximal parabolic and optimal elliptic regularity of divergence-form operators.

M. Heida, S. Nesenenko, Stochastic homogenization of rate-dependent models of monotone type in plasticity, Asymptotic Analysis, 112 (2019), pp. 185--212, DOI 10.3233/ASY-181502 .AbstractIn this work we deal with the stochastic homogenization of the initial boundary value problems of monotone type. The models of monotone type under consideration describe the deformation behaviour of inelastic materials with a microstructure which can be characterised by random measures. Based on the Fitzpatrick function concept we reduce the study of the asymptotic behaviour of monotone operators associated with our models to the problem of the stochastic homogenization of convex functionals within an ergodic and stationary setting. The concept of Fitzpatrick's function helps us to introduce and show the existence of the weak solutions for rate-dependent systems. The derivations of the homogenization results presented in this work are based on the stochastic two-scale convergence in Sobolev spaces. For completeness, we also present some two-scale homogenization results for convex functionals, which are related to the classical Gamma-convergence theory.

A. Bacho, E. Emmrich, A. Mielke, An existence result and evolutionary Gamma-convergence for perturbed gradient systems, Journal of Evolution Equations, 19 (2019), pp. 479--522, DOI 10.1007/s00028-019-00484-x .AbstractWe consider the initial-value problem for the perturbed gradient flows, where a differential inclusion is formulated in terms of a subdifferential of an energy functional, a subdifferential of a dissipation potential and a more general perturbation, which is assumed to be continuous and to satisfy a suitable growth condition. Under additional assumptions on the dissipation potential and the energy functional, existence of strong solutions is shown by proving convergence of a semi-implicit discretization scheme with a variational approximation technique.

P. Colli, G. Gilardi, J. Sprekels, A distributed control problem for a fractional tumor growth model, Mathematics - Open Access Journal, 7 (2019), pp. 792/1--792/32, DOI 10.3390/math7090792 .AbstractIn this paper, we study the distributed optimal control of a system of three evolutionary equations involving fractional powers of three selfadjoint, monotone, unbounded linear operators having compact resolvents. The system is a generalization of a Cahn--Hilliard type phase field system modeling tumor growth that goes back to Hawkins-Daarud et al. (Int. J. Numer. Math. Biomed. Eng. 28(2012), 3--24.) The aim of the control process, which could be realized by either administering a drug or monitoring the nutrition, is to keep the tumor cell fraction under control while avoiding possible harm for the patient. In contrast to previous studies, in which the occurring unbounded operators governing the diffusional regimes were all given by the Laplacian with zero Neumann boundary conditions, the operators may in our case be different; more generally, we consider systems with fractional powers of the type that were studied in the recent work Adv. Math. Sci. Appl.28(2019), 343--375 by the present authors. In our analysis, we show the Fréchet differentiability of the associated control-to-state operator, establish the existence of solutions to the associated adjoint system, and derive the first-order necessary conditions of optimality for a cost functional of tracking type.

P. Colli, G. Gilardi, J. Sprekels, Deep quench approximation and optimal control of general Cahn--Hilliard systems with fractional operators and double obstacle potentials, Discrete and Continuous Dynamical Systems -- Series S, published online on 21.12.2019, urlhttps://doi.org/10.3934/dcdss.2020213, DOI 10.3934/dcdss.2020213 .AbstractIn the recent paper ”Well-posedness and regularity for a generalized fractional Cahn--Hilliard system”, the same authors derived general well-posedness and regularity results for a rather general system of evolutionary operator equations having the structure of a Cahn--Hilliard system. The operators appearing in the system equations were fractional versions in the spectral sense of general linear operators A and B having compact resolvents and are densely defined, unbounded, selfadjoint, and monotone in a Hilbert space of functions defined in a smooth domain. The associated double-well potentials driving the phase separation process modeled by the Cahn--Hilliard system could be of a very general type that includes standard physically meaningful cases such as polynomial, logarithmic, and double obstacle nonlinearities. In the subsequent paper ”Optimal distributed control of a generalized fractional Cahn--Hilliard system” (Appl. Math. Optim. (2018), https://doi.org/10.1007/s00245-018-9540-7) by the same authors, an analysis of distributed optimal control problems was performed for such evolutionary systems, where only the differentiable case of certain polynomial and logarithmic double-well potentials could be admitted. Results concerning existence of optimizers and first-order necessary optimality conditions were derived, where more restrictive conditions on the operators A and B had to be assumed in order to be able to show differentiability properties for the associated control-to-state operator. In the present paper, we complement these results by studying a distributed control problem for such evolutionary systems in the case of nondifferentiable nonlinearities of double obstacle type. For such nonlinearities, it is well known that the standard constraint qualifications cannot be applied to construct appropriate Lagrange multipliers. To overcome this difficulty, we follow here the so-called ”deep quench” method. This technique, in which the nondifferentiable double obstacle nonlinearity is approximated by differentiable logarithmic nonlinearities, was first developed by P. Colli, M.H. Farshbaf-Shaker and J. Sprekels in the paper ”A deep quench approach to the optimal control of an Allen--Cahn equation with dynamic boundary conditions and double obstacles” (Appl. Math. Optim. 71 (2015), pp. 1-24) and has proved to be a powerful tool in a number of optimal control problems with double obstacle potentials in the framework of systems of Cahn--Hilliard type. We first give a general convergence analysis of the deep quench approximation that includes an error estimate and then demonstrate that its use leads in the double obstacle case to appropriate first-order necessary optimality conditions in terms of a variational inequality and the associated adjoint state system.

P. Colli, G. Gilardi, J. Sprekels, Well-posedness and regularity for a generalized fractional Cahn--Hilliard system, Rendiconti Lincei -- Matematica e Applicazioni, 30 (2019), pp. 437--478.AbstractIn this paper, we investigate a rather general system of two operator equations that has the structure of a viscous or nonviscous Cahn--Hilliard system in which nonlinearities of double-well type occur. Standard cases like regular or logarithmic potentials, as well as non-differentiable potentials involving indicator functions, are admitted. The operators appearing in the system equations are fractional versions of general linear operators A and B, where the latter are densely defined, unbounded, self-adjoint and monotone in a Hilbert space of functions defined in a smooth domain and have compact resolvents. In this connection, we remark the fact that our definition of the fractional power of operators uses the approach via spectral theory. Typical cases are given by standard second-order elliptic differential operators (e.g., the Laplacian) with zero Dirichlet or Neumann boundary conditions, but also other cases like fourth-order systems or systems involving the Stokes operator are covered by the theory. We derive in this paper general well-posedness and regularity results that extend corresponding results which are known for either the non-fractional Laplacian with zero Neumann boundary condition or the fractional Laplacian with zero Dirichlet condition. These results are entirely new if at least one of the operators A and B differs from the Laplacian. It turns out that the first eigenvalue λ1 of A plays an important und not entirely obvious role: if λ1 is positive, then the operators A and B may be completely unrelated; if, however, λ1 equals zero, then it must be simple and the corresponding one-dimensional eigenspace has to consist of the constant functions and to be a subset of the domain of definition of a certain fractional power of B. We are able to show general existence, uniqueness, and regularity results for both these cases, as well as for both the viscous and the nonviscous system.

P. Colli, A. Signori, J. Sprekels, Optimal control of a phase field system modelling tumor growth with chemotaxis and singular potentials, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, pp. published online on 21.10.2019, urlhttps://doi.org/10.1007/s00245-019-09618-6, DOI 10.1007/s00245-019-09618-6 .AbstractA distributed optimal control problem for an extended model of phase field type for tumor growth is addressed. In this model, the chemotaxis effects are also taken into account. The control is realized by two control variables that design the dispensation of some drugs to the patient. The cost functional is of tracking type, whereas the potential setting has been kept quite general in order to allow regular and singular potentials to be considered. In this direction, some relaxation terms have been introduced in the system. We show the well-posedness of the state system, the Fréchet differentiability of the control-to-state operator in a suitable functional analytic framework, and, lastly, we characterize the first-order necessary conditions of optimality in terms of a variational inequality involving the adjoint variables.

P. Dondl, Th. Frenzel, A. Mielke, A gradient system with a wiggly energy and relaxed EDP-convergence, ESAIM. Control, Optimisation and Calculus of Variations, 25 (2019), pp. 68/1--68/45, DOI 10.1051/cocv/2018058 .AbstractIf gradient systems depend on a microstructure, we want to derive a macroscopic gradient structure describing the effective behavior of the microscopic system. We introduce a notion of evolutionary Gamma-convergence that relates the microscopic energy and the microscopic dissipation potential with their macroscopic limits via Gamma-convergence. We call this notion relaxed EDP-convergence since the special structure of the dissipation functional may not be preserved under Gamma-convergence. However, by investigating the kinetic relation we derive the macroscopic dissipation potential.

E. Emmrich, R. Lasarzik, Existence of weak solutions to a dynamic model for smectic-A liquid crystals under undulations, IMA Journal of Applied Mathematics, 84 (2019), pp. 1143--1176, DOI 10.1093/imamat/hxz030 .AbstractA nonlinear model due to Soddemann et al. [37] and Stewart [38] describing incompressible smectic-A liquid crystals under flow is studied. In comparison to previously considered models, this particular model takes into account possible undulations of the layers away from equilibrium, which has been observed in experiments. The emerging decoupling of the director and the layer normal is incorporated by an additional evolution equation for the director. Global existence of weak solutions to this model is proved via a Galerkin approximation with eigenfunctions of the associated linear differential operators in the three-dimensional case.

S.P. Frigeri, C.G. Gal, M. Grasselli, J. Sprekels, Strong solutions to nonlocal 2D Cahn--Hilliard--Navier--Stokes systems with nonconstant viscosity, degenerate mobility and singular potential, Nonlinearity, 32 (2019), pp. 678--727, DOI 10.1088/1361-6544/aaedd0 .AbstractWe consider a nonlinear system which consists of the incompressible Navier-Stokes equations coupled with a convective nonlocal Cahn-Hilliard equation. This is a diffuse interface model which describes the motion of an incompressible isothermal mixture of two (partially) immiscible fluids having the same density. We suppose that the viscosity depends smoothly on the order parameter as well as the mobility. Moreover, we assume that the mobility is degenerate at the pure phases and that the potential is singular (e.g. of logarithmic type). This system is endowed with no-slip boundary condition for the (average) velocity and homogeneous Neumann boundary condition for the chemical potential. Thus the total mass is conserved. In the two-dimensional case, this problem was already analyzed in some joint papers of the first three authors. However, in the present general case, only the existence of a global weak solution, the (conditional) weak-strong uniqueness and the existence of the global attractor were proven. Here we are able to establish the existence of a (unique) strong solution through an approximation procedure based on time discretization. As a consequence, we can prove suitable uniform estimates which allow us to show some smoothness of the global attractor. Finally, we discuss the existence of strong solutions for the convective nonlocal Cahn-Hilliard equation, with a given velocity field, in the three dimensional case as well.

G. Gilardi, J. Sprekels, Asymptotic limits and optimal control for the Cahn--Hilliard system with convection and dynamic boundary conditions, Nonlinear Analysis. An International Mathematical Journal, 178 (2019), pp. 1--31, DOI 10.1016/j.na.2018.07.007 .AbstractIn this paper, we study initial-boundary value problems for the Cahn--Hilliard system with convection and nonconvex potential, where dynamic boundary conditions are assumed for both the associated order parameter and the corresponding chemical potential. While recent works addressed the case of viscous Cahn--Hilliard systems, the `pure' nonviscous case is investigated here. In its first part, the paper deals with the asymptotic behavior of the solutions as time approaches infinity. It is shown that the $omega$-limit of any trajectory can be characterized in terms of stationary solutions, provided the initial data are sufficiently smooth. The second part of the paper deals with the optimal control of the system by the fluid velocity. Results concerning existence and first-order necessary optimality conditions are proved. Here, we have to restrict ourselves to the case of everywhere defined smooth potentials. In both parts of the paper, we start from corresponding known results for the viscous case, derive sufficiently strong estimates that are uniform with respect to the (positive) viscosity parameter, and then let the viscosity tend to zero to establish the sought results for the nonviscous case.

V. Laschos, A. Mielke, Geometric properties of cones with applications on the Hellinger--Kantorovich space, and a new distance on the space of probability measures, Journal of Functional Analysis, 276 (2019), pp. 3529--3576, DOI 10.1016/j.jfa.2018.12.013 .AbstractBy studying general geometric properties of cone spaces, we prove the existence of a distance on the space of Probability measures that turns the Hellinger--Kantorovich space into a cone space over the space of probabilities measures. Here we exploit a natural two-parameter scaling property of the Hellinger-Kantorovich distance. For the new space, we obtain a full characterization of the geodesics. We also provide new geometric properties for the original space, including a two-parameter rescaling and reparametrization of the geodesics, local-angle condition and some partial K-semiconcavity of the squared distance, that it will be used in a future paper to prove existence of gradient flows.

J. Sprekels, H. Wu, Optimal distributed control of a Cahn--Hilliard--Darcy system with mass sources, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, pp. published online on 24.01.2019, urlhttps://doi.org/10.1007/s00245-019-09555-4, DOI 10.1007/s00245-019-09555-4 .AbstractIn this paper, we study an optimal control problem for a two-dimensional Cahn--Hilliard--Darcy system with mass sources that arises in the modeling of tumor growth. The aim is to monitor the tumor fraction in a finite time interval in such a way that both the tumor fraction, measured in terms of a tracking type cost functional, is kept under control and minimal harm is inflicted to the patient by administering the control, which could either be a drug or nutrition. We first prove that the optimal control problem admits a solution. Then we show that the control-to-state operator is Fréchet differentiable between suitable Banach spaces and derive the first-order necessary optimality conditions in terms of the adjoint variables and the usual variational inequality.

A.F.M. TERElst, J. Rehberg, Consistent operator semigroups and their interpolation, Journal of Operator Theory, 82 (2019), pp. 3--21, DOI 10.7900/jot.2017nov15.2233 .AbstractUnder a mild regularity condition we prove that the generator of the interpolation of two C0-semigroups is the interpolation of the two generators.

R. Lasarzik, Approximation and optimal control of dissipative solutions to the Ericksen--Leslie system, Numerical Functional Analysis and Optimization. An International Journal, 40 (2019), pp. 1721--1767, DOI 10.1080/01630563.2019.1632895 .AbstractWe analyze the Ericksen--Leslie system equipped with the Oseen--Frank energy in three space dimensions. Recently, the author introduced the concept of dissipative solutions. These solutions show several advantages in comparison to the earlier introduced measure-valued solutions. In this article, we argue that dissipative solutions can be numerically approximated by a relative simple scheme, which fulfills the norm-restriction on the director in every step. We introduce a semi-discrete scheme and derive an approximated version of the relative-energy inequality for solutions of this scheme. Passing to the limit in the semi-discretization, we attain dissipative solutions. Additionally, we introduce an optimal control scheme, show the existence of an optimal control and a possible approximation strategy. We prove that the cost functional is lower semi-continuous with respect to the convergence of this approximation and argue that an optimal control is attained in the case that there exists a solution admitting additional regularity.

R. Lasarzik, Measure-valued solutions to the Ericksen--Leslie model equipped with the Oseen--Frank energy, Nonlinear Analysis. An International Mathematical Journal, 179 (2019), pp. 146--183, DOI 10.1016/j.na.2018.08.013 .AbstractIn this article, we prove the existence of measure-valued solutions to the Ericksen-Leslie system equipped with the Oseen--Frank energy. We introduce the concept of generalized gradient Young measures. Via a Galerkin approximation, we show the existence of weak solutions to a regularized system and attain measure-valued solutions for vanishing regularization. Additionally, it is shown that the measure-valued solution fulfills an energy inequality.

R. Lasarzik, Weak-strong uniqueness for measure-valued solutions to the Ericksen--Leslie model equipped with the Oseen--Frank free energy, Journal of Mathematical Analysis and Applications, 470 (2019), pp. 36--90, DOI 10.1016/j.jmaa.2018.09.051 .AbstractWe analyze the Ericksen-Leslie system equipped with the Oseen--Frank energy in three space dimensions. Recently, the author introduced the concept of measure-valued solutions to this system and showed the global existence of these generalized solutions. In this paper, we show that suitable measure-valued solutions, which fulfill an associated energy inequality, enjoy the weak-strong uniqueness property, i.e. the measure-valued solution agrees with a strong solution if the latter exists. The weak-strong uniqueness is shown by a relative energy inequality for the associated nonconvex energy functional.

M. Liero, S. Melchionna, The weighted energy-dissipation principle and evolutionary Gamma-convergence for doubly nonlinear problems, ESAIM. Control, Optimisation and Calculus of Variations, 25 (2019), pp. 36/1--36/38, DOI 10.1051/cocv/2018023 .AbstractWe consider a family of doubly nonlinear evolution equations that is given by families of convex dissipation potentials, nonconvex energy functionals, and external forces parametrized by a small parameter ε. For each of these problems, we introduce the so-called weighted energy-dissipation (WED) functional, whose minimizer correspond to solutions of an elliptic-in-time regularization of the target problems with regularization parameter δ. We investigate the relation between the Γ-convergence of the WED functionals and evolutionary Γ-convergence of the associated systems. More precisely, we deal with the limits δ→0, ε→0, as well as δ+ ε→0 either in the sense of Γ-convergence of functionals or in the sense of evolutionary Γ-convergence of functional-driven evolution problems, or both. Additionally, we provide some quantitative estimates on the rate of convergence for the limit ε→0, in the case of quadratic dissipation potentials and uniformly λ-convex energy functionals. Finally, we discuss a homogenization problem as an example of application.

A. Alphonse, Ch.M. Elliott, J. Terra, A coupled ligand-receptor bulk-surface system on a moving domain: Well posedness, regularity and convergence to equilibrium, SIAM Journal on Mathematical Analysis, 50 (2018), pp. 1544--1592, DOI 10.1137/16M110808X .AbstractWe prove existence, uniqueness, and regularity for a reaction-diffusion system of coupled bulk-surface equations on a moving domain modelling receptor-ligand dynamics in cells. The nonlinear coupling between the three unknowns is through the Robin boundary condition for the bulk quantity and the right hand sides of the two surface equations. Our results are new even in the non-moving setting, and in this case we also show exponential convergence to a steady state. The primary complications in the analysis are indeed the nonlinear coupling and the Robin boundary condition. For the well posedness and essential boundedness of solutions we use several De Giorgi-type arguments, and we also develop some useful estimates to allow us to apply a Steklov averaging technique for time-dependent operators to prove that solutions are strong. Some of these auxiliary results presented in this paper are of independent interest by themselves.

M. Heida, M. Röger, Large deviation principle for a stochastic Allen--Cahn equation, Journal of Theoretical Probability, 31 (2018), pp. 364--401, DOI 10.1007/s10959-016-0711-7 .AbstractThe Allen-Cahn equation is a prototype model for phase separation processes, a fundamental example of a nonlinear spatial dynamic and an important approximation of a geometric evolution equation by a reaction-diffusion equation. Stochastic perturbations, especially in the case of additive noise, to the Allen-Cahn equation have attracted considerable attention. We consider here an alternative random perturbation determined by a Brownian flow of spatial diffeomorphism that was introduced by Röger and Weber [Stoch. Partial Differ. Equ. Anal. Comput. 1 (2013)]. We first provide a large deviation principle for stochastic flows in spaces of functions that are Hölder-continuous in time, which extends results by Budhiraja, Dupuis and Maroulas [Ann. Probab. 36 (2008)]. From this result and a continuity argument we deduce a large deviation principle for the Allen-Cahn equation perturbed by a Brownian flow in the limit of small noise. Finally, we present two asymptotic reductions of the large deviation functional.

M. Heida, On convergences of the squareroot approximation scheme to the Fokker--Planck operator, Mathematical Models & Methods in Applied Sciences, 28 (2018), pp. 2599--2635, DOI 10.1142/S0218202518500562 .AbstractWe study the qualitative convergence properties of a finite volume scheme that recently was proposed by Lie, Fackeldey and Weber [SIAM Journal on Matrix Analysis and Applications 2013 (34/2)] in the context of conformation dynamics. The scheme was derived from physical principles and is called the squareroot approximation (SQRA) scheme. We show that solutions to the SQRA equation converge to solutions of the Fokker-Planck equation using a discrete notion of G-convergence. Hence the squareroot approximation turns out to be a usefull approximation scheme to the Fokker-Planck equation in high dimensional spaces. As an example, in the special case of stationary Voronoi tessellations we use stochastic two-scale convergence to prove that this setting satisfies the G-convergence property. In particular, the class of tessellations for which the G-convergence result holds is not trivial.

E. MecaÁlvarez, A. Münch, B. Wagner, Localized instabilities and spinodal decomposition in driven systems in the presence of elasticity, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 97 (2018), pp. 012801/1--012801/12, DOI 10.1103/PhysRevE.97.012801 .AbstractWe study numerically and analytically the instabilities associated with phase separation in a solid layer on which an external material flux is imposed. The first instability is localized within a boundary layer at the exposed free surface by a process akin to spinodal decomposition. In the limiting static case, when there is no material flux, the coherent spinodal decomposition is recovered. In the present problem stability analysis of the time-dependent and non-uniform base states as well as numerical simulations of the full governing equations are used to establish the dependence of the wavelength and onset of the instability on parameter settings and its transient nature as the patterns eventually coarsen into a flat moving front. The second instability is related to the Mullins-Sekerka instability in the presence of elasticity and arises at the moving front between the two phases when the flux is reversed. Stability analyses of the full model and the corresponding sharp-interface model are carried out and compared. Our results demonstrate how interface and bulk instabilities can be analysed within the same framework which allows to identify and distinguish each of them clearly. The relevance for a detailed understanding of both instabilities and their interconnections in a realistic setting are demonstrated for a system of equations modelling the lithiation/delithiation processes within the context of Lithium ion batteries.

A. Ceretani, C.N. Rautenberg, The Boussinesq system with mixed non-smooth boundary conditions and ``do-nothing'' boundary flow, Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 70 (2019), pp. 14/1--14/24 (published online on 07.12.2018), DOI 10.1007/s00033-018-1058-y .AbstractA stationary Boussinesq system for an incompressible viscous fluid in a bounded domain with a nontrivial condition at an open boundary is studied. We consider a novel non-smooth boundary condition associated to the heat transfer on the open boundary that involves the temperature at the boundary, the velocity of the fluid, and the outside temperature. We show that this condition is compatible with two approaches at dealing with the do-nothing boundary condition for the fluid: 1) the directional do-nothing condition and 2) the do-nothing condition together with an integral bound for the backflow. Well-posedness of variational formulations is proved for each problem.

M. Becker, Th. Frenzel, Th. Niedermeyer, S. Reichelt, A. Mielke, M. Bär, Local control of globally competing patterns in coupled Swift--Hohenberg equations, Chaos. An Interdisciplinary Journal of Nonlinear Science, 28 (2018), pp. 043121/1--043121/11, DOI 10.1063/1.5018139 .AbstractWe present analytical and numerical investigations of two anti-symmetrically coupled 1D Swift--Hohenberg equations (SHEs) with cubic nonlinearities. The SHE provides a generic formulation for pattern formation at a characteristic length scale. A linear stability analysis of the homogeneous state reveals a wave instability in addition to the usual Turing instability of uncoupled SHEs. We performed weakly nonlinear analysis in the vicinity of the codimension-two point of the Turing-wave instability, resulting in a set of coupled amplitude equations for the Turing pattern as well as left and right traveling waves. In particular, these complex Ginzburg--Landau-type equations predict two major things: there exists a parameter regime where multiple different patterns are stable with respect to each other; and that the amplitudes of different patterns interact by local mutual suppression. In consequence, different patterns can coexist in distinct spatial regions, separated by localized interfaces. We identified specific mechanisms for controlling the position of these interfaces, which distinguish what kinds of patterns the interface connects and thus allow for global pattern selection. Extensive simulations of the original SHEs confirm our results.

D. Belomestny, J.G.M. Schoenmakers, Projected particle methods for solving McKean--Vlasov equations, SIAM Journal on Numerical Analysis, 56 (2018), pp. 3169--3195, DOI 10.1137/17M1111024 .AbstractWe propose a novel projection-based particle method for solving McKean--Vlasov stochastic differential equations. Our approach is based on a projection-type estimation of the marginal density of the solution in each time step. The projection-based particle method leads in many situations to a significant reduction of numerical complexity compared to the widely used kernel density estimation algorithms. We derive strong convergence rates and rates of density estimation. The convergence analysis, particularly in the case of linearly growing coefficients, turns out to be rather challenging and requires some new type of averaging technique. This case is exemplified by explicit solutions to a class of McKean--Vlasov equations with affine drift. The performance of the proposed algorithm is illustrated by several numerical examples.

P. Colli, G. Gilardi, J. Sprekels, On a Cahn--Hilliard system with convection and dynamic boundary conditions, Annali di Matematica Pura ed Applicata. Serie Quarta. Fondazione Annali di Matematica Pura ed Applicata, c/o Dipartimento di Matematica ``U. Dini'', Firenze; Springer-Verlag, Heidelberg. English, French, German, Italian, English abstracts., 197 (2018), pp. 1445--1475, DOI 10.1007/s10231-018-0732-1 .AbstractThis paper deals with an initial and boundary value problem for a system coupling equation and boundary condition both of Cahn--Hilliard type; an additional convective term with a forced velocity field, which could act as a control on the system, is also present in the equation. Either regular or singular potentials are admitted in the bulk and on the boundary. Both the viscous and pure Cahn--Hilliard cases are investigated, and a number of results is proven about existence of solutions, uniqueness, regularity, continuous dependence, uniform boundedness of solutions, strict separation property. A complete approximation of the problem, based on the regularization of maximal monotone graphs and the use of a Faedo--Galerkin scheme, is introduced and rigorously discussed.

P. Colli, G. Gilardi, J. Sprekels, On the longtime behavior of a viscous Cahn--Hilliard system with convection and dynamic boundary conditions, Journal of Elliptic and Parabolic Equations, 4 (2018), pp. 327--347, DOI 10.1007/s41808-018-0021-6 .AbstractIn this paper, we study the longtime asymptotic behavior of a phase separation process occurring in a three-dimensional domain containing a fluid flow of given velocity. This process is modeled by a viscous convective Cahn--Hilliard system, which consists of two nonlinearly coupled second-order partial differential equations for the unknown quantities, the chemical potential and an order parameter representing the scaled density of one of the phases. In contrast to other contributions, in which zero Neumann boundary conditions were are assumed for both the chemical potential and the order parameter, we consider the case of dynamic boundary conditions, which model the situation when another phase transition takes place on the boundary. The phase transition processes in the bulk and on the boundary are driven by free energies functionals that may be nondifferentiable and have derivatives only in the sense of (possibly set-valued) subdifferentials. For the resulting initial-boundary value system of Cahn--Hilliard type, general well-posedness results have been established in piera recent contribution by the same authors. In the present paper, we investigate the asymptotic behavior of the solutions as times approaches infinity. More precisely, we study the ω-limit (in a suitable topology) of every solution trajectory. Under the assumptions that the viscosity coefficients are strictly positive and that at least one of the underlying free energies is differentiable, we prove that the omegalimit is meaningful and that all of its elements are solutions to the corresponding stationary system, where the component representing the chemical potential is a constant.

P. Colli, G. Gilardi, J. Sprekels, Optimal distributed control of a generalized fractional Cahn--Hilliard system, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, pp. published online on 15.11.2018, urlhttps://doi.org/10.1007/s00245-018-9540-7, DOI 10.1007/s00245-018-9540-7 .AbstractIn the recent paper “Well-posedness and regularity for a generalized fractional Cahn--Hilliard system” by the same authors, general well-posedness results have been established for a class of evolutionary systems of two equations having the structure of a viscous Cahn--Hilliard system, in which nonlinearities of double-well type occur. The operators appearing in the system equations are fractional versions in the spectral sense of general linear operators A,B, having compact resolvents, which are densely defined, unbounded, selfadjoint, and monotone in a Hilbert space of functions defined in a smooth domain. In this work we complement the results given in quoted paper by studying a distributed control problem for this evolutionary system. The main difficulty in the analysis is to establish a rigorous Fréchet differentiability result for the associated control-to-state mapping. This seems only to be possible if the state stays bounded, which, in turn, makes it necessary to postulate an additional global boundedness assumption. One typical situation, in which this assumption is satisfied, arises when B is the negative Laplacian with zero Dirichlet boundary conditions and the nonlinearity is smooth with polynomial growth of at most order four. Also a case with logarithmic nonlinearity can be handled. Under the global boundedness assumption, we establish existence and first-order necessary optimality conditions for the optimal control problem in terms of a variational inequality and the associated adjoint state system.

P. Gurevich, S. Reichelt, Pulses in FitzHugh--Nagumo systems with rapidly oscillating coefficients, Multiscale Modeling & Simulation. A SIAM Interdisciplinary Journal, 16 (2018), pp. 833--856.AbstractThis paper is devoted to pulse solutions in FitzHugh-Nagumo systems that are coupled parabolic equations with rapidly periodically oscillating coefficients. In the limit of vanishing periods, there arises a two-scale FitzHugh-Nagumo system, which qualitatively and quantitatively captures the dynamics of the original system. We prove existence and stability of pulses in the limit system and show their proximity on any finite time interval to pulse-like solutions of the original system.

J. Haskovec, S. Hittmeir, P. Markowich, A. Mielke, Decay to equilibrium for energy-reaction-diffusion systems, SIAM Journal on Mathematical Analysis, 50 (2018), pp. 1037--1075, DOI 10.1137/16M1062065 .AbstractWe derive thermodynamically consistent models of reaction-diffusion equations coupled to a heat equation. While the total energy is conserved, the total entropy serves as a driving functional such that the full coupled system is a gradient flow. The novelty of the approach is the Onsager structure, which is the dual form of a gradient system, and the formulation in terms of the densities and the internal energy. In these variables it is possible to assume that the entropy density is strictly concave such that there is a unique maximizer (thermodynamical equilibrium) given linear constraints on the total energy and suitable density constraints. We consider two particular systems of this type, namely, a diffusion-reaction bipolar energy transport system, and a drift-diffusion-reaction energy transport system with confining potential. We prove corresponding entropy-entropy production inequalities with explicitely calculable constants and establish the convergence to thermodynamical equilibrium, at first in entropy and further in L1 using Cziszar-Kullback-Pinsker type inequalities.

G. Lazzaroni, R. Rossi, M. Thomas, R. Toader, Rate-independent damage in thermo-viscoelastic materials with inertia, Journal of Dynamics and Differential Equations, 30 (2018), pp. 1311--1364, DOI 10.1007/s10884-018-9666-y .AbstractWe present a model for rate-independent, unidirectional, partial damage in visco-elastic materials with inertia and thermal effects. The damage process is modeled by means of an internal variable, governed by a rate-independent flow rule. The heat equation and the momentum balance for the displacements are coupled in a highly nonlinear way. Our assumptions on the corresponding energy functional also comprise the case of the Ambrosio-Tortorelli phase-field model (without passage to the brittle limit). We discuss a suitable weak formulation and prove an existence theorem obtained with the aid of a (partially) decoupled time-discrete scheme and variational convergence methods. We also carry out the asymptotic analysis for vanishing viscosity and inertia and obtain a fully rate-independent limit model for displacements and damage, which is independent of temperature.

A. Muntean, S. Reichelt, Corrector estimates for a thermo-diffusion model with weak thermal coupling, Multiscale Modeling & Simulation. A SIAM Interdisciplinary Journal, 16 (2018), pp. 807--832, DOI 10.1137/16M109538X .AbstractThe present work deals with the derivation of corrector estimates for the two-scale homogenization of a thermo-diffusion model with weak thermal coupling posed in a heterogeneous medium endowed with periodically arranged high-contrast microstructures. The terminology “weak thermal coupling” refers here to the variable scaling in terms of the small homogenization parameter ε of the heat conduction-diffusion interaction terms, while the “high-contrast” is thought particularly in terms of the heat conduction properties of the composite material. As main target, we justify the first-order terms of the multiscale asymptotic expansions in the presence of coupled fluxes, induced by the joint contribution of Sorret and Dufour-like effects. The contrasting heat conduction combined with cross coupling lead to the main mathematical difficulty in the system. Our approach relies on the method of periodic unfolding combined with ε-independent estimates for the thermal and concentration fields and for their coupled fluxes

M. Thomas, C. Bilgen, K. Weinberg, Phase-field fracture at finite strains based on modified invariants: A note on its analysis and simulations, GAMM-Mitteilungen, 40 (2018), pp. 207--237, DOI 10.1002/gamm.201730004 .AbstractPhase-field models have already been proven to predict complex fracture patterns in two and three dimensions for brittle fracture at small strains. In this paper we discuss a model for phase-field fracture at finite deformations in more detail. Among the identification of crack location and projection of crack growth the numerical stability is one of the main challenges in solid mechanics. We here present a phase-field model at finite strains, which takes into account the anisotropy of damage by applying an anisotropic split and the modified invariants of the right Cauchy-Green strain tensor. We introduce a suitable weak notion of solution that also allows for a spatial and temporal discretization of the model. In this framework we study the existence of solutions %Second the mathematical background of the approach is examined and and we show that the time-discrete solutions converge in a weak sense to a solution of the time-continuous formulation of the model. Numerical examples in two and three space dimensions are carried out in the range of validity of the analytical results.

R. Lasarzik, Dissipative solution to the Ericksen--Leslie system equipped with the Oseen--Frank energy, Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, pp. published online on 29.11.2018, urlhttps://doi.org/10.1007/s00033-018-1053-3, DOI 10.1007/s00033-018-1053-3 .AbstractWe analyze the Ericksen-Leslie system equipped with the Oseen?Frank energy in three space dimensions. The new concept of dissipative solutions is introduced. Recently, the author introduced the concept of measure-valued solutions to the considered system and showed global existence as well as weak-strong uniqueness of these generalized solutions. In this paper, we show that the expectation of the measure valued solution is a dissipative solution. The concept of a dissipative solution itself relies on an inequality instead of an equality, but is described by functions instead of parametrized measures. These solutions exist globally and fulfill the weak-strong uniqueness property. Additionally, we generalize the relative energy inequality to solutions fulfilling different nonhomogeneous Dirichlet boundary conditions and incorporate the influence of a temporarily constant electromagnetic field. Relying on this generalized energy inequality, we investigate the long-time behavior and show that all solutions converge for the large time limit to a certain steady state.

M. Liero, S. Melchionna, The weighted energy-dissipation principle and evolutionary Gamma-convergence for doubly nonlinear problems, ESAIM. Control, Optimisation and Calculus of Variations, 25 (2019), pp. 36/1--36/38, DOI 10.1051/cocv/2018023 .AbstractWe consider a family of doubly nonlinear evolution equations that is given by families of convex dissipation potentials, nonconvex energy functionals, and external forces parametrized by a small parameter ε. For each of these problems, we introduce the so-called weighted energy-dissipation (WED) functional, whose minimizer correspond to solutions of an elliptic-in-time regularization of the target problems with regularization parameter δ. We investigate the relation between the Γ-convergence of the WED functionals and evolutionary Γ-convergence of the associated systems. More precisely, we deal with the limits δ→0, ε→0, as well as δ+ ε→0 either in the sense of Γ-convergence of functionals or in the sense of evolutionary Γ-convergence of functional-driven evolution problems, or both. Additionally, we provide some quantitative estimates on the rate of convergence for the limit ε→0, in the case of quadratic dissipation potentials and uniformly λ-convex energy functionals. Finally, we discuss a homogenization problem as an example of application.

M. Liero, S. Reichelt, Homogenization of Cahn--Hilliard-type equations via evolutionary Gamma-convergence, NoDEA. Nonlinear Differential Equations and Applications, 25 (2018), pp. 6/1--6/31, DOI 10.1007/s00030-018-0495-9 .AbstractIn this paper we discuss two approaches to evolutionary Γ-convergence of gradient systems in Hilbert spaces. The formulation of the gradient system is based on two functionals, namely the energy functional and the dissipation potential, which allows us to employ Γ-convergence methods. In the first approach we consider families of uniformly convex energy functionals such that the limit passage of the time-dependent problems can be based on the theory of evolutionary variational inequalities as developed by Daneri and Savaré 2010. The second approach uses the equivalent formulation of the gradient system via the energy-dissipation principle and follows the ideas of Sandier and Serfaty 2004. We apply both approaches to rigorously derive homogenization limits for Cahn-Hilliard-type equations. Using the method of weak and strong two-scale convergence via periodic unfolding, we show that the energy and dissipation functionals Γ-converge. In conclusion, we will give specific examples for the applicability of each of the two approaches.

S. Bergmann, D.A. Barragan-Yani, E. Flegel, K. Albe, B. Wagner, Anisotropic solid-liquid interface kinetics in silicon: An atomistically informed phase-field model, Modelling and Simulation in Materials Science and Engineering, 25 (2017), pp. 065015/1--065015/20, DOI 10.1088/1361-651X/aa7862 .AbstractWe present an atomistically informed parametrization of a phase-field model for describing the anisotropic mobility of liquid-solid interfaces in silicon. The model is derived from a consistent set of atomistic data and thus allows to directly link molecular dynamics and phase field simulations. Expressions for the free energy density, the interfacial energy and the temperature and orientation dependent interface mobility are systematically fitted to data from molecular dynamics simulations based on the Stillinger-Weber interatomic potential. The temperature-dependent interface velocity follows a Vogel-Fulcher type behavior and allows to properly account for the dynamics in the undercooled melt.

K. Disser, A.F.M. TERElst, J. Rehberg, On maximal parabolic regularity for non-autonomous parabolic operators, Journal of Differential Equations, 262 (2017), pp. 2039--2072.AbstractWe consider linear inhomogeneous non-autonomous parabolic problems associated to sesquilinear forms, with discontinuous dependence of time. We show that for these problems, the property of maximal parabolic regularity can be extrapolated to time integrability exponents r ≠ 2. This allows us to prove maximal parabolic Lr-regularity for discontinuous non-autonomous second-order divergence form operators in very general geometric settings and to prove existence results for related quasilinear equations.

K. Disser, J. Rehberg, A.F.M. TERElst, Hölder estimates for parabolic operators on domains with rough boundary, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V, XVII (2017), pp. 65--79.AbstractIn this paper we investigate linear parabolic, second-order boundary value problems with mixed boundary conditions on rough domains. Assuming only boundedness/ellipticity on the coefficient function and very mild conditions on the geometry of the domain -- including a very weak compatibility condition between the Dirichlet boundary part and its complement -- we prove Hölder continuity of the solution in space and time.

S. Reichelt, Corrector estimates for a class of imperfect transmission problems, Asymptotic Analysis, 105 (2017), pp. 3--26, DOI 10.3233/ASY-171432 .AbstractBased on previous homogenization results for imperfect transmission problems in two-component domains with periodic microstructure, we derive quantitative estimates for the difference between the microscopic and macroscopic solution. This difference is of order ερ, where ε > 0 describes the periodicity of the microstructure and ρ ∈ (0 , ½] depends on the transmission condition at the interface between the two components. The corrector estimates are proved without assuming additional regularity for the local correctors using the periodic unfolding method.

M. Heida, A. Mielke, Averaging of time-periodic dissipation potentials in rate-independent processes, Discrete and Continuous Dynamical Systems -- Series S, 10 (2017), pp. 1303--1327.AbstractWe study the existence and well-posedness of rate-independent systems (or hysteresis operators) with a dissipation potential that oscillates in time with period ε. In particular, for the case of quadratic energies in a Hilbert space, we study the averaging limit ε→0 and show that the effctive dissipation potential is given by the minimum of all friction thresholds in one period, more precisely as the intersection of all the characteristic domains. We show that the rates of the process do not converge weakly, hence our analysis uses the notion of energetic solutions and relies on a detailed estimates to obtain a suitable equi-continuity of the solutions in the limit ε→0.

M. Heida, Stochastic homogenization of rate-independent systems, Continuum Mechanics and Thermodynamics, 29 (2017), pp. 853--894, DOI 10.1007/s00161-017-0564-z .AbstractWe study the stochastic and periodic homogenization 1-homogeneous convex functionals. We proof some convergence results with respect to stochastic two-scale convergence, which are related to classical Gamma-convergence results. The main result is a general liminf-estimate for a sequence of 1-homogeneous functionals and a two-scale stability result for sequences of convex sets. We apply our results to the homogenization of rateindependent systems with 1-homogeneous dissipation potentials and quadratic energies. In these applications, both the energy and the dissipation potential have an underlying stochastic microscopic structure. We study the particular homogenization problems of Prandlt-Reuss plasticity, Coulomb friction on a macroscopic surface and Coulomb friction on microscopic fissure.

M. Liero, A. Mielke, M.A. Peletier, D.R.M. Renger, On microscopic origins of generalized gradient structures, Discrete and Continuous Dynamical Systems -- Series S, 10 (2017), pp. 1--35, DOI 10.3934/dcdss.2017001 .AbstractClassical gradient systems have a linear relation between rates and driving forces. In generalized gradient systems we allow for arbitrary relations derived from general non-quadratic dissipation potentials. This paper describes two natural origins for these structures. A first microscopic origin of generalized gradient structures is given by the theory of large-deviation principles. While Markovian diffusion processes lead to classical gradient structures, Poissonian jump processes give rise to cosh-type dissipation potentials. A second origin arises via a new form of convergence, that we call EDP-convergence. Even when starting with classical gradient systems, where the dissipation potential is a quadratic functional of the rate, we may obtain a generalized gradient system in the evolutionary Gamma-limit. As examples we treat (i) the limit of a diffusion equation having a thin layer of low diffusivity, which leads to a membrane model, and (ii) the limit of diffusion over a high barrier, which gives a reaction-diffusion system.

R. Rossi, M. Thomas, Coupling rate-independent and rate-dependent processes: Existence results, SIAM Journal on Mathematical Analysis, 49 (2017), pp. 1419--1494.AbstractWe address the analysis of an abstract system coupling a rate-independet process with a second order (in time) nonlinear evolution equation. We propose suitable weak solution concepts and obtain existence results by passing to the limit in carefully devised time-discretization schemes. Our arguments combine techniques from the theory of gradient systems with the toolbox for rate-independent evolution, thus reflecting the mixed character of the problem. Finally, we discuss applications to a class of rate-independent processes in visco-elastic solids with inertia, and to a recently proposed model for damage with plasticity.

R. Rossi, M. Thomas, From adhesive to brittle delamination in visco-elastodynamics, Mathematical Models & Methods in Applied Sciences, 27 (2017), pp. 1489--1546, DOI 10.1142/S0218202517500257 .AbstractIn this paper we analyze a system for brittle delamination between two visco-elastic bodies, also subject to inertia, which can be interpreted as a model for dynamic fracture. The rate-independent flow rule for the delamination parameter is coupled with the momentum balance for the displacement, including inertia. This model features a nonsmooth constraint ensuring the continuity of the displacements outside the crack set, which is marked by the support of the delamination parameter. A weak solvability concept, generalizing the notion of energetic solution for rate-independent systems to the present mixed rate-dependent/rate-independent frame, is proposed. Via refined variational convergence techniques, existence of solutions is proved by passing to the limit in approximating systems which regularize the nonsmooth constraint by conditions for adhesive contact. The presence of the inertial term requires the design of suitable recovery spaces small enough to provide compactness but large enough to recover the information on the crack set in the limit.

P. Colli, G. Gilardi, E. Rocca, J. Sprekels, Optimal distributed control of a diffuse interface model of tumor growth, Nonlinearity, 30 (2017), pp. 2518--2546.AbstractIn this paper, a distributed optimal control problem is studied for a diffuse interface model of tumor growth which was proposed by Hawkins--Daruud et al. in citeHZO. The model consists of a Cahn-Hilliard equation for the tumor cell fraction $vp$ coupled to a reaction-diffusion equation for a function $s$ representing the nutrient-rich extracellular water volume fraction. The distributed control $u$ monitors as a right-hand side the equation for $s$ and can be interpreted as a nutrient supply or a medication, while the cost function, which is of standard tracking type, is meant to keep the tumor cell fraction under control during the evolution. We show that the control-to-state operator is Fréchet differentiable between appropriate Banach spaces and derive the first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables.

P. Colli, G. Gilardi, E. Rocca, J. Sprekels, Asymptotic analyses and error estimates for a Cahn--Hilliard type phase field system modelling tumor growth, Discrete and Continuous Dynamical Systems, 10 (2017), pp. 37--54.AbstractThis paper is concerned with a phase field system of Cahn--Hilliard type that is related to a tumor growth model and consists of three equations in gianni terms of the variables order parameter, chemical potential and nutrient concentration. This system has been investigated in the recent papers citeCGH and citeCGRS gianni from the viewpoint of well-posedness, long time bhv and asymptotic convergence as two positive viscosity coefficients tend to zero at the same time. Here, we continue the analysis performed in citeCGRS by showing two independent sets of results as just one of the coefficents tends to zero, the other remaining fixed. We prove convergence results, uniqueness of solutions to the two resulting limit problems, and suitable error estimates

P. Colli, G. Gilardi, J. Sprekels, Global existence for a nonstandard viscous Cahn--Hilliard system with dynamic boundary condition, SIAM Journal on Mathematical Analysis, 49 (2017), pp. 1732--1760, DOI 10.1137/16M1087539 .AbstractIn this paper, we study a model for phase segregation taking place in a spatial domain that was introduced by Podio-Guidugli in Ric. Mat. 55 (2006), pp. 105-118. The model consists of a strongly coupled system of nonlinear parabolic differential equations, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically. In contrast to the existing literature about this PDE system, we consider here a dynamic boundary condition involving the Laplace-Beltrami operator for the order parameter. This boundary condition models an additional nonconserving phase transition occurring on the surface of the domain. Different well-posedness results are shown, depending on the smoothness properties of the involved bulk and surface free energies.

P. Krejčí, E. Rocca, J. Sprekels, Unsaturated deformable porous media flow with thermal phase transition, Mathematical Models & Methods in Applied Sciences, 27 (2017), pp. 2675--2710, DOI 10.1142/S0218202517500555 .AbstractIn the present paper, a continuum model is introduced for fluid flow in a deformable porous medium, where the fluid may undergo phase transitions. Typically, such problems arise in modeling liquid-solid phase transformations in groundwater flows. The system of equations is derived here from the conservation principles for mass, momentum, and energy and from the Clausius-Duhem inequality for entropy. It couples the evolution of the displacement in the matrix material, of the capillary pressure, of the absolute temperature, and of the phase fraction. Mathematical results are proved under the additional hypothesis that inertia effects and shear stresses can be neglected. For the resulting highly nonlinear system of two PDEs, one ODE and one ordinary differential inclusion with natural initial and boundary conditions, existence of global in time solutions is proved by means of cut-off techniques and suitable Moser-type estimates.

H. Meinlschmidt, Ch. Meyer, J. Rehberg, Optimal control of the thermistor problem in three spatial dimensions. Part 1: Existence of optimal solutions, SIAM Journal on Control and Optimization, 55 (2017), pp. 2876--2904, DOI 10.1137/16M1072644 .AbstractThis paper is concerned with the state-constrained optimal control of the three-dimensional thermistor problem, a fully quasilinear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Local existence, uniqueness and continuity for the state system are derived by employing maximal parabolic regularity in the fundamental theorem of Prüss. Global solutions are addressed, which includes analysis of the linearized state system via maximal parabolic regularity, and existence of optimal controls is shown if the temperature gradient is under control. The adjoint system involving measures is investigated using a duality argument. These results allow to derive first-order necessary conditions for the optimal control problem in form of a qualified optimality system. The theoretical findings are illustrated by numerical results.

H. Meinlschmidt, Ch. Meyer, J. Rehberg, Optimal control of the thermistor problem in three spatial dimensions. Part 2: Optimality conditions, SIAM Journal on Control and Optimization, 55 (2017), pp. 2368--2392, DOI 10.1137/16M1072656 .AbstractThis paper is concerned with the state-constrained optimal control of the three-dimensional thermistor problem, a fully quasilinear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Local existence, uniqueness and continuity for the state system are derived by employing maximal parabolic regularity in the fundamental theorem of Prüss. Global solutions are addressed, which includes analysis of the linearized state system via maximal parabolic regularity, and existence of optimal controls is shown if the temperature gradient is under control. The adjoint system involving measures is investigated using a duality argument. These results allow to derive first-order necessary conditions for the optimal control problem in form of a qualified optimality system. The theoretical findings are illustrated by numerical results.

A. Roggensack, Ch. Kraus, Existence of weak solutions for the Cahn--Hilliard reaction model including elastic effects and damage, Journal of Partial Differential Equations, 30 (2017), pp. 111-145, DOI 10.4208/jpde.v30.n2.2 .AbstractIn this paper, we introduce and study analytically a vectorial Cahn-Hilliard reaction model coupled with rate-dependent damage processes. The recently proposed Cahn-Hilliard reaction model can e.g. be used to describe the behavior of electrodes of lithium-ion batteries as it includes both the intercalation reactions at the surfaces and the separation into different phases. The coupling with the damage process allows considering simultaneously the evolution of a damage field, a second important physical effect occurring during the charging or discharging of lithium-ion batteries. Mathematically, this is realized by a Cahn-Larché system with a non-linear Newton boundary condition for the chemical potential and a doubly non-linear differential inclusion for the damage evolution. We show that this system possesses an underlying generalized gradient structure which incorporates the non-linear Newton boundary condition. Using this gradient structure and techniques from the field of convex analysis we are able to prove constructively the existence of weak solutions of the coupled PDE system.

J. Sprekels, E. Valdinoci, A new type of identification problems: Optimizing the fractional order in a nonlocal evolution equation, SIAM Journal on Control and Optimization, 55 (2017), pp. 70--93.AbstractIn this paper, we consider a rather general linear evolution equation of fractional type, namely a diffusion type problem in which the diffusion operator is the power of a positive definite operator having a positive and discrete spectrum. We prove existence, uniqueness and differentiability properties with respect to the fractional parameter. These results are then employed to derive existence as well as first-order necessary and second-order sufficient optimality conditions for a minimization problem, which is inspired by considerations in mathematical biology. In this problem, the fractional parameter $s$ serves as the “control parameter” that needs to be chosen in such a way as to minimize a given cost functional. This problem constitutes a new classof identification problems: while usually in identification problems the type of the differential operator is prescribed and one or several of its coefficient functions need to be identified, in the present case one has to determine the type of the differential operator itself. This problem exhibits the inherent analytical difficulty that with changing fractional parameter also the domain of definition, and thus the underlying function space, of the fractional operator changes.

A. Glitzky, M. Liero, Analysis of p(x)-Laplace thermistor models describing the electrothermal behavior of organic semiconductor devices, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 34 (2017), pp. 536--562.AbstractWe study a stationary thermistor model describing the electrothermal behavior of organic semiconductor devices featuring non-Ohmic current-voltage laws and self-heating effects. The coupled system consists of the current-flow equation for the electrostatic potential and the heat equation with Joule heating term as source. The self-heating in the device is modeled by an Arrhenius-like temperature dependency of the electrical conductivity. Moreover, the non-Ohmic electrical behavior is modeled by a power law such that the electrical conductivity depends nonlinearly on the electric field. Notably, we allow for functional substructures with different power laws, which gives rise to a $p(x)$-Laplace-type problem with piecewise constant exponent. We prove the existence and boundedness of solutions in the two-dimensional case. The crucial point is to establish the higher integrability of the gradient of the electrostatic potential to tackle the Joule heating term. The proof of the improved regularity is based on Caccioppoli-type estimates, Poincaré inequalities, and a Gehring-type Lemma for the $p(x)$-Laplacian. Finally, Schauder's fixed-point theorem is used to show the existence of solutions.

M. Thomas, Ch. Zanini, Cohesive zone-type delamination in visco-elasticity, Discrete and Continuous Dynamical Systems -- Series S, 10 (2017), pp. 1487--1517, DOI 10.3934/dcdss.2017077 .AbstractWe study a model for the rate-independent evolution of cohesive zone delamination in a visco-elastic solid, also exposed to dynamics effects. The main feature of this model, inspired by [Ortiz&Pandoli99Int.J.Numer.Meth.Eng.], is that the surface energy related to the crack opening depends on the history of the crack separation between the two sides of the crack path, and allows for different responses upon loading and unloading.

Due to the presence of multivalued and unbounded operators featuring non-penetration and the `memory'-constraint in the strong formulation of the problem, we prove existence of a weaker notion of solution, known as semistable energetic solution, pioneered in [Roubicek09M2AS] and refined in [Rossi&Thomas15WIAS-Preprint2113].

N. Ahmed, T.Ch. Rebollo, V. John, S. Rubino, A review of variational multiscale methods for the simulation of turbulent incompressible flows, Archives of Computational Methods in Engineering. State of the Art Reviews, 24 (2017), pp. 115--164.AbstractVarious realizations of variational multiscale (VMS) methods for simulating turbulent incompressible flows have been proposed in the past fifteen years. All of these realizations obey the basic principles of VMS methods: They are based on the variational formulation of the incompressible Navier--Stokes equations and the scale separation is defined by projections. However, apart from these common basic features, the various VMS methods look quite different. In this review, the derivation of the different VMS methods is presented in some detail and their relation among each other and also to other discretizations is discussed. Another emphasis consists in giving an overview about known results from the numerical analysis of the VMS methods. A few results are presented in detail to highlight the used mathematical tools. Furthermore, the literature presenting numerical studies with the VMS methods is surveyed and the obtained results are summarized.

N. Ahmed, T.Ch. Rebollo, V. John, S. Rubino, Analysis of a full space-time discretization of the Navier--Stokes equations by a local projection stabilization method, IMA Journal of Numerical Analysis, 37 (2017), pp. 1437--1467, DOI 10.1093/imanum/drw048 .AbstractA finite element error analysis of a local projection stabilization (LPS) method for the time-dependent Navier--Stokes equations is presented. The focus is on the high-order term-by-term stabilization method that has one level, in the sense that it is defined on a single mesh, and in which the projection-stabilized structure of standard LPS methods is replaced by an interpolation-stabilized structure. The main contribution is on proving, theoretically and numerically, the optimal convergence order of the arising fully discrete scheme. In addition, the asymptotic energy balance is obtained for slightly smooth flows. Numerical studies support the analytical results and illustrate the potential of the method for the simulation of turbulent flows. Smooth unsteady flows are simulated with optimal order of accuracy.

N. Ahmed, On the grad-div stabilization for the steady Oseen and Navier--Stokes equations, Calcolo. A Quarterly on Numerical Analysis and Theory of Computation, 54 (2017), pp. 471--501, DOI 10.1007/s10092-016-0194-z .AbstractThis paper studies the parameter choice in the grad-div stabilization applied to the generalized problems of Oseen type. Stabilization parameters based on minimizing the H1(Ω) error of the velocity are derived which do not depend on the viscosity parameter. For the proposed parameter choices, the H1(Ω) error of the velocity is derived that shows a direct dependence on the viscosity parameter. Differences and common features to the situation for the Stokes equations are discussed. Numerical studies are presented which confirm the theoretical results. Moreover, for the Navier- Stokes equations, numerical simulations were performed on a two-dimensional ow past a circular cylinder. It turns out, for the MINI element, that the best results can be obtained without grad-div stabilization.

M. Hintermüller, C.N. Rautenberg, On the uniqueness and numerical approximation of solutions to certain parabolic quasi-variational inequalities, Portugaliae Mathematica. A Journal of the Portuguese Mathematical Society, 74 (2017), pp. 1--35.AbstractA class of abstract nonlinear evolution quasi-variational inequality (QVI) problems in function space is considered. The abstract framework developed in this paper includes constraint sets of obstacle and gradient type. The paper address the existence, uniqueness and approximation of solutions when the constraint set mapping is of a special form. Uniqueness is addressed through contractive behavior of a nonlinear mapping whose fixed points are solutions to the QVI. An axiomatic semi-discrete approximation scheme is developed, which is proven to be convergent and which is numerically implemented. The paper ends by a report on numerical tests for several nonlinear constraints of gradient-type.

A. Mielke, M. Mittnenzweig, Convergence to equilibrium in energy-reaction-diffusion systems using vector-valued functional inequalities, Journal of Nonlinear Science, 28 (2018), pp. 765--806 (published online on 11.11.2017), DOI 10.1007/s00332-017-9427-9 .AbstractWe discuss how the recently developed energy-dissipation methods for reactiondi usion systems can be generalized to the non-isothermal case. For this we use concave entropies in terms of the densities of the species and the internal energy, where the importance is that the equilibrium densities may depend on the internal energy. Using the log-Sobolev estimate and variants for lower-order entropies as well as estimates for the entropy production of the nonlinear reactions we give two methods to estimate the relative entropy by the total entropy production, namely a somewhat restrictive convexity method, which provides explicit decay rates, and a very general, but weaker compactness method.

A. Mielke, C. Patz, Uniform asymptotic expansions for the infinite harmonic chain, Zeitschrift fur Analysis und ihre Anwendungen. Journal for Analysis and its Applications, 36 (2017), pp. 437--475, DOI 10.4171/ZAA/1596 .AbstractWe study the dispersive behavior of waves in linear oscillator chains. We show that for general general dispersions it is possible to construct an expansion such that the remainder can be estimated by $1/t$ uniformly in space. In particalur we give precise asymptotics for the transition from the $1/t^1/2$ decay of nondegenerate wave numbers to the generate $1/t^1/3$ decay of generate wave numbers. This involves a careful description of the oscillatory integral involving the Airy function.

A. Mielke, R.I.A. Patterson, M.A. Peletier, D.R.M. Renger, Non-equilibrium thermodynamical principles for chemical reactions with mass-action kinetics, SIAM Journal on Applied Mathematics, 77 (2017), pp. 1562--1585, DOI 10.1137/16M1102240 .AbstractWe study stochastic interacting particle systems that model chemical reaction networks on the micro scale, converging to the macroscopic Reaction Rate Equation. One abstraction level higher, we study the ensemble of such particle systems, converging to the corresponding Liouville transport equation. For both systems, we calculate the corresponding large deviations and show that under the condition of detailed balance, the large deviations induce a non-linear relation between thermodynamic fluxes and free energy driving force.

E. Cinti, J. Davila, M. DelPino, Solutions of the fractional Allen--Cahn equation which are invariant under screw motion, Journal of the London Mathematical Society. Second Series, 94 (2016), pp. 295--313.AbstractWe establish existence and non-existence results for entire solutions to the fractional Allen--Cahn equation in R3 , which vanish on helicoids and are invariant under screw-motion. In addition, we prove that helicoids are surfaces with vanishing nonlocal mean curvature.

K. Disser, G.P. Galdi, G. Mazzone, P. Zunino, Inertial motions of a rigid body with a cavity filled with a viscous liquid, Archive for Rational Mechanics and Analysis, 221 (2016), pp. 487--526.AbstractWe consider the system of equations modeling the free motion of a rigid body with a cavity filled by a viscous (Navier-Stokes) liquid. Zhukovskiy's Theorem states that in the limit of time going to infinity, the relative fluid velocity tends to 0 and the rigid velocity of the full structure tends to a steady rotation around one of the principle axes of inertia. We give a rigorous proof of this result. In particular, we prove that every global weak solution in a suitable class is subject to Zhukovskiy's Theorem, and note that existence of these solutions has been established. Independently of the geometry and of parameters, this shows that the presence of fluid prevents precession of the body in the limit. In general, we cannot predict which axis will be attained, but we can show stability of the largest axis and provide criteria on the initial data which are decisive in special cases.

M. Cozzi, A. Farina, E. Valdinoci, Monotonicity formulae and classification results for singular, degenerate, anisotropic PDEs, Advances in Mathematics, 293 (2016), pp. 343--381.AbstractWe consider possibly degenerate and singular elliptic equations in a possibly anisotropic medium. We obtain monotonicity results for the energy density, rigidity results for the solutions and classi?cation results for the singularity/degeneracy/anisotropy allowed. As far as we know, these results are new even in the case of non-singular and non- degenerate anisotropic equations.

S.P. Frigeri, Global existence of weak solutions for a nonlocal model for two-phase flows of incompressible fluids with unmatched densities, Mathematical Models & Methods in Applied Sciences, 26 (2016), pp. 1957--1993.AbstractWe consider a diffuse interface model for an incompressible isothermal mixture of two viscous Newtonian fluids with different densities in a bounded domain in two or three space dimensions. The model is the nonlocal version of the one recently derived by Abels, Garcke and Grün and consists of a Navier-Stokes type system coupled with a convective nonlocal Cahn-Hilliard equation. The density of the mixture depends on an order parameter. For this nonlocal system we prove existence of global dissipative weak solutions for the case of singular double-well potentials and non degenerate mobilities. To this goal we devise an approach which is completely independent of the one employed by Abels, Depner and Garcke to establish existence of weak solutions for the local Abels et al. model.

M. Liero, A. Mielke, G. Savaré, Optimal transport in competition with reaction: The Hellinger--Kantorovich distance and geodesic curves, SIAM Journal on Mathematical Analysis, 48 (2016), pp. 2869--2911.AbstractWe discuss a new notion of distance on the space of finite and nonnegative measures on Ω ⊂ ℝ d, which we call Hellinger-Kantorovich distance. It can be seen as an inf-convolution of the well-known Kantorovich-Wasserstein distance and the Hellinger-Kakutani distance. The new distance is based on a dynamical formulation given by an Onsager operator that is the sum of a Wasserstein diffusion part and an additional reaction part describing the generation and absorption of mass. We present a full characterization of the distance and some of its properties. In particular, the distance can be equivalently described by an optimal transport problem on the cone space over the underlying space Ω. We give a construction of geodesic curves and discuss examples and their general properties.

D. Peschka, N. Rotundo, M. Thomas, Towards doping optimization of semiconductor lasers, Journal of Computational and Theoretical Transport, 45 (2016), pp. 410--423.AbstractWe discuss analytical and numerical methods for the optimization of optoelectronic devices by performing optimal control of the PDE governing the carrier transport with respect to the doping profile. First, we provide a cost functional that is a sum of a regularization and a contribution, which is motivated by the modal net gain that appears in optoelectronic models of bulk or quantum-well lasers. Then, we state a numerical discretization, for which we study optimized solutions for different regularizations and for vanishing weights.

M. Dai, E. Feireisl, E. Rocca, G. Schimperna, M.E. Schonbek, On asymptotic isotropy for a hydrodynamic model of liquid crystals, Asymptotic Analysis, 97 (2016), pp. 189--210.AbstractWe study a PDE system describing the motion of liquid crystals by means of the Q?tensor description for the crystals coupled with the incompressible Navier-Stokes system. Using the method of Fourier splitting, we show that solutions of the system tend to the isotropic state at the rate (1 + t)?? as t ? ? 1 for a certain ? > 2 .

S. Dipierro, O. Savin, E. Valdinoci, Graph properties for nonlocal minimal surfaces, Calculus of Variations and Partial Differential Equations, 55 (2016), pp. 86/1--86/25.AbstractIn this paper we show that a nonlocal minimal surface which is a graph outside a cylinder is in fact a graph in the whole of the space. As a consequence, in dimension 3, we show that the graph is smooth. The proofs rely on convolution techniques and appropriate integral estimates which show the pointwise validity of an Euler?Lagrange equation related to the nonlocal mean curvature.

S. Patrizi, E. Valdinoci, Relaxation times for atom dislocations in crystals, Calculus of Variations and Partial Differential Equations, 55 (2016), pp. 71/1--71/44.AbstractWe study the relaxation times for a parabolic differential equation whose solution represents the atom dislocation in a crystal. The equation that we consider comprises the classical Peierls?Nabarro model as a particular case, and it allows also long range interactions. It is known that the dislocation function of such a model has the tendency to concentrate at single points, which evolve in time according to the external stress and a singular, long range potential. Depending on the orientation of the dislocation function at these points, the potential may be either attractive or repulsive, hence collisions may occur in the latter case and, at the collision time, the dislocation function does not disappear. The goal of this paper is to provide accurate estimates on the relaxation times of the system after collision. More precisely, we take into account the case of two and three colliding points, and we show that, after a small transition time subsequent to the collision, the dislocation function relaxes exponentially fast to a steady state. In this sense, the system exhibits two different decay behaviors, namely an exponential time decay versus a polynomial decay in the space variables (and these two homogeneities are kept separate during the time evolution).

P. Bringmann, C. Carstensen, Ch. Merdon, Guaranteed error control for the pseudostress approximation of the Stokes equations, Numerical Methods for Partial Differential Equations. An International Journal, 32 (2016), pp. 1411--1432.AbstractThe pseudostress approximation of the Stokes equations rewrites the stationary Stokes equations with pure (but possibly inhomogeneous) Dirichlet boundary conditions as another (equivalent) mixed scheme based on a stress in H(div) and the velocity in $L^2$. Any standard mixed finite element function space can be utilized for this mixed formulation, e.g. the Raviart-Thomas discretization which is related to the Crouzeix-Raviart nonconforming finite element scheme in the lowest-order case. The effective and guaranteed a posteriori error control for this nonconforming velocity-oriented discretization can be generalized to the error control of some piecewise quadratic velocity approximation that is related to the discrete pseudostress. The analysis allows for local inf-sup constants which can be chosen in a global partition to improve the estimation. Numerical examples provide strong evidence for an effective and guaranteed error control with very small overestimation factors even for domains with large anisotropy.

M. Bulíček, A. Glitzky, M. Liero, Systems describing electrothermal effects with p(x)-Laplacian like structure for discontinuous variable exponents, SIAM Journal on Mathematical Analysis, 48 (2016), pp. 3496--3514.AbstractWe consider a coupled system of two elliptic PDEs, where the elliptic term in the first equation shares the properties of the p(x)-Laplacian with discontinuous exponent, while in the second equation we have to deal with an a priori L1 term on the right hand side. Such a system of equations is suitable for the description of various electrothermal effects, in particular those, where the non-Ohmic behavior can change dramatically with respect to the spatial variable. We prove the existence of a weak solution under very weak assumptions on the data and also under general structural assumptions on the constitutive equations of the model. The main difficulty consists in the fact that we have to overcome simultaneously two obstacles - the discontinuous variable exponent (which limits the use of standard methods) and the L1 right hand side of the heat equation. Our existence proof based on Galerkin approximation is highly constructive and therefore seems to be suitable also for numerical purposes.

P. Colli, G. Gilardi, J. Sprekels, A boundary control problem for the viscous Cahn--Hilliard equation with dynamic boundary conditions, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 73 (2016), pp. 195--225, DOI 10.1007/s00245-015-9299-z .AbstractA boundary control problem for the viscous Cahn-Hilliard equations with possibly singular potentials and dynamic boundary conditions is studied and first order necessary conditions for optimality are proved.

P. Colli, G. Gilardi, J. Sprekels, Constrained evolution for a quasilinear parabolic equation, Journal of Optimization Theory and Applications, 170 (2016), pp. 713--734.AbstractIn the present contribution, a feedback control law is studied for a quasilinear parabolic equation. First, we prove the well-posedness and some regularity results for the Cauchy--Neumann problem for this equation, modified by adding an extra term which is a multiple of the subdifferential of the distance function from a closed convex set K of L2(Ω). Then, we consider convex sets of obstacle or double-obstacle type, and we can act on the factor of the feedback control in order to be able to reach the convex set within a finite time, by proving rigorously this property.

P. Colli, G. Gilardi, J. Sprekels, Distributed optimal control of a nonstandard nonlocal phase field system, AIMS Mathematics, 1 (2016), pp. 246--281.AbstractWe investigate a distributed optimal control problem for a nonlocal phase field model of viscous Cahn-Hilliard type. The model constitutes a nonlocal version of a model for two-species phase segregation on an atomic lattice under the presence of diffusion that has been studied in a series of papers by P. Podio-Guidugli and the present authors. The model consists of a highly nonlinear parabolic equation coupled to an ordinary differential equation. The latter equation contains both nonlocal and singular terms that render the analysis difficult. Standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional are of standard type. We show that the problem admits a solution, and we derive the first-order necessary conditions of optimality.

P. Colli, G. Gilardi, J. Sprekels, On an application of Tikhonov's fixed point theorem to a nonlocal Cahn--Hilliard type system modeling phase separation, Journal of Differential Equations, 260 (2016), pp. 7940--7964.AbstractThis paper investigates a nonlocal version of a model for phase separation on an atomic lattice that was introduced by P. Podio-Guidugli in Ric. Mat.55 (2006) 105-118. The model consists of an initial-boundary value problem for a nonlinearly coupled system of two partial differential equations governing the evolution of an order parameter ρ and the chemical potential μ. Singular contributions to the local free energy in the form of logarithmic or double-obstacle potentials are admitted. In contrast to the local model, which was studied by P. Podio-Guidugli and the present authors in a series of recent publications, in the nonlocal case the equation governing the evolution of the order parameter contains in place of the Laplacian a nonlocal expression that originates from nonlocal contributions to the free energy and accounts for possible long-range interactions between the atoms. It is shown that just as in the local case the model equations are well posed, where the technique of proving existence is entirely different: it is based on an application of Tikhonov's fixed point theorem in a rather unusual separable and reflexive Banach space.

J. Dávila, M. DelPino, S. Dipierro, E. Valdinoci, Nonlocal Delaunay surfaces, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 137 (2016), pp. 357--380.AbstractWe construct codimension 11 surfaces of any dimension that minimize a periodic nonlocal perimeter functional among surfaces that are periodic, cylindrically symmetric and decreasing. These surfaces may be seen as a nonlocal analogue of the classical Delaunay surfaces (onduloids). For small volume, most of their mass tends to be concentrated in a periodic array and the surfaces are close to a periodic array of balls (in fact, we give explicit quantitative bounds on these facts).

A. Farina, E. Valdinoci, 1D symmetry for semilinear PDEs from the limit interface of the solution, Communications in Partial Differential Equations, 41 (2016), pp. 665--682.AbstractWe study bounded, monotone solutions of ?u = W?(u) in the whole of ?n, where W is a double-well potential. We prove that under suitable assumptions on the limit interface and on the energy growth, u is 1D. In particular, differently from the previous literature, the solution is not assumed to have minimal properties and the cases studied lie outside the range of ?-convergence methods. We think that this approach could be fruitful in concrete situations, where one can observe the phase separation at a large scale and wishes to deduce the values of the state parameter in the vicinity of the interface. As a simple example of the results obtained with this point of view, we mention that monotone solutions with energy bounds, whose limit interface does not contain a vertical line through the origin, are 1D, at least up to dimension 4.

H. Meinlschmidt, J. Rehberg, Hölder-estimates for non-autonomous parabolic problems with rough data, Evolution Equations and Control Theory, 5 (2016), pp. 147--184.AbstractIn this paper we establish Hölder estimates for solutions to non-autonomous parabolic equations on non-smooth domains which are complemented with mixed boundary conditions. The corresponding elliptic operators are of divergence type, the coefficient matrix of which depends only measurably on time. These results are in the tradition of the classical book of Ladyshenskaya et al., which also serves as the starting point for our investigations.

X. Ros-Oton, E. Valdinoci, The Dirichlet problem for nonlocal operators with kernels: Convex and nonconvex domains, Advances in Mathematics, 288 (2016), pp. 732--790.AbstractWe study the interior regularity of solutions to a Dirichlet problem for anisotropic operators of fractional type. A prototype example is given by the sum of one-dimensional fractional Laplacians in fixed, given directions. We prove here that an interior differentiable regularity theory holds in convex domains. When the spectral measure is a bounded function and the domain is smooth, the same regularity theory applies. In particular, solutions always possess a classical first derivative. The assumptions on the domain are sharp, since if the domain is not convex and the spectral measure is singular, we construct an explicit counterexample.

S. Reichelt, Error estimates for elliptic equations with not exactly periodic coefficients, Advances in Mathematical Sciences and Applications, 25 (2016), pp. 117--131.AbstractThis note is devoted to the derivation of quantitative estimates for linear elliptic equations with coefficients that are not exactly ε-periodic and the ellipticity constant may degenerate for vanishing ε. Here ε>0 denotes the ratio between the microscopic and the macroscopic length scale. It is shown that for degenerating and non-degenerating coefficients the error between the original solution and the effective solution is of order √ε. Therefore suitable test functions are constructed via the periodic unfolding method and a gradient folding operator making only minimal additional assumptions on the given data and the effective solution with respect to the macroscopic scale.

N. Ahmed, G. Matthies, Numerical study of SUPG and LPS methods combined with higher order variational time discretization schemes applied to time-dependent convection-diffusion-reaction equations, Journal of Scientific Computing, 67 (2016), pp. 988--1018.AbstractThis paper considers the numerical solution of time-dependent convection-diffusion-reaction equations. We shall employ combinations of streamline-upwind Petrov-Galerkin (SUPG) and local projection stabilization (LPS) methods in space with the higher order variational time discretization schemes. In particular, we consider time discretizations by discontinuous Galerkin (dG) methods and continuous Galerkin-Petrov (cGP) methods. Several numerical tests have been performed to assess the accuracy of combinations of spatial and temporal discretization schemes. Furthermore, the dependence of the results on the stabilization parameters of the spatial discretizations are discussed. Finally the long-time behavior of overshoots and undershoots is investigated.

M.H. FarshbafShaker, C. Hecht, Optimal control of elastic vector-valued Allen--Cahn variational inequalities, SIAM Journal on Control and Optimization, 54 (2016), pp. 129--152.AbstractIn this paper we consider a elastic vector-valued Allen--Cahn MPCC (Mathematical Programs with Complementarity Constraints) problem. We use a regularization approach to get the optimality system for the subproblems. By passing to the limit in the optimality conditions for the regularized subproblems, we derive certain generalized first-order necessary optimality conditions for the original problem.

S.P. Frigeri, E. Rocca, J. Sprekels, Optimal distributed control of a nonlocal Cahn--Hilliard/Navier--Stokes system in two dimensions, SIAM Journal on Control and Optimization, 54 (2016), pp. 221 -- 250.AbstractWe study a diffuse interface model for incompressible isothermal mixtures of two immiscible fluids coupling the Navier-Stokes system with a convective nonlocal Cahn-Hilliard equation in two dimensions of space. We apply recently proved well-posedness and regularity results in order to establish existence of optimal controls as well as first-order necessary optimality conditions for an associated optimal control problem in which a distributed control is applied to the fluid flow.

A. Mielke, M.A. Peletier, D.R.M. Renger, A generalization of Onsager's reciprocity relations to gradient flows with nonlinear mobility, Journal of Non-Equilibrium Thermodynamics, 41 (2016), pp. 141--149.AbstractOnsager's 1931 `reciprocity relations' result connects microscopic time-reversibility with a symmetry property of corresponding macroscopic evolution equations. Among the many consequences is a variational characterization of the macroscopic evolution equation as a gradient-flow, steepest-ascent, or maximal-entropy-production equation. Onsager's original theorem is limited to close-to-equilibrium situations, with a Gaussian invariant measure and a linear macroscopic evolution. In this paper we generalize this result beyond these limitations, and show how the microscopic time-reversibility leads to natural generalized symmetry conditions, which take the form of generalized gradient flows.

A. Mielke, T. Roubíček, Rate-independent elastoplasticity at finite strains and its numerical approximation, Mathematical Models & Methods in Applied Sciences, 26 (2016), pp. 2203--2236.AbstractGradient plasticity at large strains with kinematic hardening is analyzed as quasistatic rate-independent evolution. The energy functional with a frame-indifferent polyconvex energy density and the dissipation are approximated numerically by finite elements and implicit time discretization, such that a computationally implementable scheme is obtained. The non-selfpenetration as well as a possible frictionless unilateral contact is considered and approximated numerically by a suitable penalization method which keeps polyconvexity and simultaneously by-passes the Lavrentiev phenomenon. The main result concerns the convergence of the numerical scheme towards energetic solutions.

In the case of incompressible plasticity and of nonsimple materials, where the energy depends on the second derivative of the deformation, we derive an explicit stability criterion for convergence relating the spatial discretization and the penalizations.

A. Mielke, R. Rossi, G. Savaré, Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems, Journal of the European Mathematical Society (JEMS), 18 (2016), pp. 2107--2165.AbstractBalanced Viscosity solutions to rate-independent systems arise as limits of regularized rate-independent ows by adding a superlinear vanishing-viscosity dissipation. We address the main issue of proving the existence of such limits for innite-dimensional systems and of characterizing them by a couple of variational properties that combine a local stability condition and a balanced energy-dissipation identity. A careful description of the jump behavior of the solutions, of their dierentiability properties, and of their equivalent representation by time rescaling is also presented. Our techniques rely on a suitable chain-rule inequality for functions of bounded variation in Banach spaces, on rened lower semicontinuity-compactness arguments, and on new BVestimates that are of independent interest.

R.I.A. Patterson, Properties of the solutions of delocalised coagulation and inception problems with outflow boundaries, Journal of Evolution Equations, 16 (2016), pp. 261--291.AbstractWell posedness is established for a family of equations modelling particle populations undergoing delocalised coagulation, advection, inflow and outflow in a externally specified velocity field. Very general particle types are allowed while the spatial domain is a bounded region of $d$-dimensional space for which every point lies on exactly one streamline associated with the velocity field. The problem is formulated as a semi-linear ODE in the Banach space of bounded measures on particle position and type space. A local Lipschitz property is established in total variation norm for the propagators (generalised semi-groups) associated with the problem and used to construct a Picard iteration that establishes local existence and global uniqueness for any initial condition. The unique weak solution is shown further to be a differentiable or at least bounded variation strong solution under smoothness assumptions on the parameters of the coagulation interaction. In the case of one spatial dimension strong differentiability is established even for coagulation parameters with a particular bounded variation structure in space. This one dimensional extension establishes the convergence of the simulation processes studied in [Patterson, Stoch. Anal. Appl. 31, 2013] to a unique and differentiable limit.

K. Disser, M. Meyries, J. Rehberg, A unified framework for parabolic equations with mixed boundary conditions and diffusion on interfaces, Journal of Mathematical Analysis and Applications, 430 (2015), pp. 1102--1123.AbstractIn this paper we consider scalar parabolic equations in a general non-smooth setting with emphasis on mixed interface and boundary conditions. In particular, we allow for dynamics and diffusion on a Lipschitz interface and on the boundary, where diffusion coefficients are only assumed to be bounded, measurable and positive semidefinite. In the bulk, we additionally take into account diffusion coefficients which may degenerate towards a Lipschitz surface. For this problem class, we introduce a unified functional analytic framework based on sesquilinear forms and show maximal regularity for the corresponding abstract Cauchy problem.

K. Disser, M. Liero, On gradient structures for Markov chains and the passage to Wasserstein gradient flows, Networks and Heterogeneous Media, 10 (2015), pp. 233-253.AbstractWe study the approximation of Wasserstein gradient structures by their finite-dimensional analog. We show that simple finite-volume discretizations of the linear Fokker-Planck equation exhibit the recently established entropic gradient-flow structure for reversible Markov chains. Then, we reprove the convergence of the discrete scheme in the limit of vanishing mesh size using only the involved gradient-flow structures. In particular, we make no use of the linearity of the equations nor of the fact that the Fokker-Planck equation is of second order.

K. Disser, H.-Chr. Kaiser, J. Rehberg, Optimal Sobolev regularity for linear second-order divergence elliptic operators occurring in real-world problems, SIAM Journal on Mathematical Analysis, 47 (2015), pp. 1719--1746.AbstractOn bounded three-dimensional domains, we consider divergence-type operators including mixed homogeneous Dirichlet and Neumann boundary conditions and discontinuous coefficient functions. We develop a geometric framework in which it is possible to prove that the operator provides an isomorphism of suitable function spaces. In particular, in these spaces, the gradient of solutions turns out to be integrable with exponent larger than the space dimension three. Relevant examples from real-world applications are provided in great detail.

S. Patrizi, E. Valdinoci, Crystal dislocations with different orientations and collisions, Archive for Rational Mechanics and Analysis, 217 (2015), pp. 231--261.AbstractWe study a parabolic differential equation whose solution represents the atom dislocation in a crystal for a general type of Peierls-Nabarro model with possibly long range interactions and an external stress. Differently from the previous literature, we treat here the case in which such dislocation is not the superpositions of transitions all occurring with the same orientations (i.e. opposite orientations are allowed as well). We show that, at a long time scale, and at a macroscopic space scale, the dislocations have the tendency to concentrate as pure jumps at points which evolve in time, driven by the external stress and by a singular potential. Due to differences in the dislocation orientations, these points may collide in finite time.

S. Patrizi, E. Valdinoci, Homogenization and Orowan's law for anisotropic fractional operators of any order, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 119 (2015), pp. 3--36.AbstractWe consider an anisotropic fractional operator and we consider the homogenization properties of an evolution equation. The scaling properties and the effective Hamiltonian that we obtain is different according to the fractional parameter. In the isotropic onedimensional case, we also prove a statement related to the so-called Orowan's law, that is an appropriate scaling of the effective Hamiltonian presents a linear behavior.

E. Rocca, R. Rossi, ``Entropic'' solutions to a thermodynamically consistent PDE system for phase transitions and damage, SIAM Journal on Mathematical Analysis, 74 (2015), pp. 2519--2586.AbstractIn this paper we analyze a PDE system modelling (non-isothermal) phase transitions and dam- age phenomena in thermoviscoelastic materials. The model is thermodynamically consistent: in particular, no small perturbation assumption is adopted, which results in the presence of quadratic terms on the right-hand side of the temperature equation, only estimated in L^1. The whole system has a highly nonlinear character. We address the existence of a weak notion of solution, referred to as “entropic”, where the temperature equation is formulated with the aid of an entropy inequality, and of a total energy inequality. This solvability concept reflects the basic principles of thermomechanics as well as the thermodynamical consistency of the model. It allows us to obtain global-in-time existence theorems without imposing any restriction on the size of the initial data. We prove our results by passing to the limit in a time discretization scheme, carefully tailored to the nonlinear features of the PDE system (with its “entropic” formulation), and of the a priori estimates performed on it. Our time-discrete analysis could be useful towards the numerical study of this model.

E. Rocca, J. Sprekels, Optimal distributed control of a nonlocal convective Cahn--Hilliard equation by the velocity in three dimensions, SIAM Journal on Control and Optimization, 53 (2015), pp. 1654--1680.AbstractIn this paper we study a distributed optimal control problem for a nonlocal convective Cahn-Hilliard equation with degenerate mobility and singular potential in three dimensions of space. While the cost functional is of standard tracking type, the control problem under investigation cannot easily be treated via standard techniques for two reasons: the state system is a highly nonlinear system of PDEs containing singular and degenerating terms, and the control variable, which is given by the velocity of the motion occurring in the convective term, is nonlinearly coupled to the state variable. The latter fact makes it necessary to state rather special regularity assumptions for the admissible controls, which, while looking a bit nonstandard, are however quite natural in the corresponding analytical framework. In fact, they are indispensable prerequisites to guarantee the well-posedness of the associated state system. In this contribution, we employ recently proved existence, uniqueness and regularity results for the solution to the associated state system in order to establish the existence of optimal controls and appropriate first-order necessary optimality conditions for the optimal control problem.

P.-É. Druet, Higher $L^p$ regularity for vector fields that satisfy divergence and rotation constraints in dual Sobolev spaces, and application to some low-frequency Maxwell equations, Discrete and Continuous Dynamical Systems, 8 (2015), pp. 479--496.AbstractWe show that Lp vector fields over a Lipschitz domain are integrable to higher exponents if their generalized divergence and rotation can be identified with bounded linear operators acting on standard Sobolev spaces. A Div-Curl Lemma-type argument provides compact embedding results for such vector fields. We investigate the regularity of the solution fields for the low-frequency approximation of the Maxwell equations in time-harmonic regime. We focus on the weak formulation 'in H' of the problem, in a reference geometrical setting allowing for material heterogeneities.

P.-É. Druet, Some mathematical problems related to the second order optimal shape of a crystallization interface, Discrete and Continuous Dynamical Systems, 35 (2015), pp. 2443--2463.AbstractWe consider the problem to optimize the stationary temperature distribution and the equilibrium shape of the solid-liquid interface in a two-phase system subject to a temperature gradient. The interface satisfies the minimization principle of the free energy, while the temperature is solving the heat equation with a radiation boundary conditions at the outer wall. Under the condition that the temperature gradient is uniformly negative in the direction of crystallization, the interface is expected to have a global graph representation. We reformulate this condition as a pointwise constraint on the gradient of the state, and we derive the first order optimality system for a class of objective functionals that account for the second surface derivatives, and for the surface temperature gradient.

S.P. Frigeri, M. Grasselli, E. Rocca, A diffuse interface model for two-phase incompressible flows with nonlocal interactions and nonconstant mobility, Nonlinearity, 28 (2015), pp. 1257--1293.AbstractWe consider a diffuse interface model for incompressible isothermal mixtures of two immiscible fluids with matched constant densities. This model consists of the Navier-Stokes system coupled with a convective nonlocal Cahn-Hilliard equation with non-constant mobility. We first prove the existence of a global weak solution in the case of non-degenerate mobilities and regular potentials of polynomial growth. Then we extend the result to degenerate mobilities and singular (e.g. logarithmic) potentials. In the latter case we also establish the existence of the global attractor in dimension two. Using a similar technique, we show that there is a global attractor for the convective nonlocal Cahn-Hilliard equation with degenerate mobility and singular potential in dimension three.

M. Heida, Existence of solutions for two types of generalized versions of the Cahn--Hilliard equation, Applications of Mathematics, 60 (2015), pp. 51--90.AbstractWe show existence of solutions to two types of generalized anisotropic Cahn-Hilliard problems: In the first case, we assume the mobility to be dependent on the concentration and its gradient, where the system is supplied with dynamic boundary conditions. In the second case, we deal with classical no-flux boundary conditions where the mobility depends on concentration u, gradient of concentration ?u and the chemical potential ?u?s?(u). The existence is shown using a newly developed generalization of gradient flows by the author and the theory of Young measures.

M. Heida, On systems of Cahn--Hilliard and Allen--Cahn equations considered as gradient flows in Hilbert spaces, Journal of Mathematical Analysis and Applications, 423 (2015), pp. 410--455.

M. Liero, Th. Koprucki, A. Fischer, R. Scholz, A. Glitzky, p-Laplace thermistor modeling of electrothermal feedback in organic semiconductors, ZAMP Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 66 (2015), pp. 2957--2977.AbstractIn large-area Organic Light-Emitting Diodes (OLEDs) spatially inhomogeneous luminance at high power due to inhomogeneous current flow and electrothermal feedback can be observed. To describe these self-heating effects in organic semiconductors we present a stationary thermistor model based on the heat equation for the temperature coupled to a p-Laplace-type equation for the electrostatic potential with mixed boundary conditions. The p-Laplacian describes the non-Ohmic electrical behavior of the organic material. Moreover, an Arrhenius-like temperature dependency of the electrical conductivity is considered. We introduce a finite-volume scheme for the system and discuss its relation to recent network models for OLEDs. In two spatial dimensions we derive a priori estimates for the temperature and the electrostatic potential and prove the existence of a weak solution by Schauder's fixed point theorem.

S. Yanchuk, L. Lücken, M. Wolfrum, A. Mielke, Spectrum and amplitude equations for scalar delay-differential equations with large delay, Discrete and Continuous Dynamical Systems, 35 (2015), pp. 537--553.AbstractThe subject of the paper are scalar delay-differential equations with large delay. Firstly, we describe the asymptotic properties of the spectrum of linear equations. Using these properties, we classify possible types of destabilization of steady states. In the limit of large delay, this classification is similar to the one for parabolic partial differential equations. We present a derivation and error estimates for amplitude equations, which describe universally the local behavior of scalar delay-differential equations close to the destabilization threshold.

S. Yanchuk, G. Giacomelli, Dynamical systems with multiple, long delayed feedbacks: Multiscale analysis and spatio-temporal equivalence, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 92 (2015), pp. 042903/1--042903/12.AbstractDynamical systems with multiple, hierarchically long delayed feedback are introduced and studied. Focusing on the phenomenological model of a Stuart-Landau oscillator with two feedbacks, we show the multiscale properties of its dynamics and demonstrate them by means of a space-time representation. For sufficiently long delays, we derive a normal form describing the system close to the destabilization. The space and temporal variables, which are involved in the space-time representation, correspond to suitable timescales of the original system. The physical meaning of the results, together with the interpretation of the description at different scales, is presented and discussed. In particular, it is shown how this representation uncovers hidden multiscale patterns such as spirals or spatiotemporal chaos. The effect of the delays size and the features of the transition between small to large delays is also analyzed. Finally, we comment on the application of the method and on its extension to an arbitrary, but finite, number of delayed feedback terms.

E. Bonetti, Ch. Heinemann, Ch. Kraus, A. Segatti, Modeling and analysis of a phase field system for damage and phase separation processes in solids, Journal of Partial Differential Equations, 258 (2015), pp. 3928--3959.AbstractIn this work, we analytically investigate a multi-component system for describing phase separation and damage processes in solids. The model consists of a parabolic diffusion equation of fourth order for the concentration coupled with an elliptic system with material dependent coefficients for the strain tensor and a doubly nonlinear differential inclusion for the damage function. The main aim of this paper is to show existence of weak solutions for the introduced model, where, in contrast to existing damage models in the literature, different elastic properties of damaged and undamaged material are regarded. To prove existence of weak solutions for the introduced system, we start with a regularized version. Then, by passing to the limit, existence results of weak solutions for the proposed model are obtained via suitable variational techniques.

S. Dipierro, E. Valdinoci, A density property for fractional weighted Sobolev spaces, Rendiconti Lincei -- Matematica e Applicazioni, 26 (2015), pp. 397--422.AbstractIn this paper we show a density property for fractional weighted Sobolev spaces. That is, we prove that any function in a fractional weighted Sobolev space can be approximated by a smooth function with compact support. The additional difficulty in this nonlocal setting is caused by the fact that the weights are not necessarily translation invariant.

S. Dipierro, E. Valdinoci, On a fractional harmonic replacement, Discrete and Continuous Dynamical Systems, 35 (2015), pp. 3377--3392.AbstractGiven $s &in;(0,1)$, we consider the problem of minimizing the Gagliardo seminorm in $H^s$ with prescribed condition outside the ball and under the further constraint of attaining zero value in a given set $K$. We investigate how the energy changes in dependence of such set. In particular, under mild regularity conditions, we show that adding a set $A$ to $K$ increases the energy of at most the measure of $A$ (this may be seen as a perturbation result for small sets $A$). Also, we point out a monotonicity feature of the energy with respect to the prescribed sets and the boundary conditions.

S. Dipierro, O. Savin, E. Valdinoci, A nonlocal free boundary problem, SIAM Journal on Mathematical Analysis, 47 (2015), pp. 4559--4605.AbstractWe consider a nonlocal free boundary problem built by a fractional Dirichlet norm plus a fractional perimeter. Among other results, we prove a monotonicity formula for the minimizers, glueing lemmata, uniform energy bounds, convergence results, a regularity theory for the planar cones and a trivialization result for the flat case. Several classical free boundary problems are limit cases of the one that we consider in this paper.

R. Rossi, M. Thomas, From an adhesive to a brittle delamination model in thermo-visco-elasticity, ESAIM. Control, Optimisation and Calculus of Variations, 21 (2015), pp. 1--59.AbstractWe address the analysis of a model for brittle delamination of two visco-elastic bodies, bonded along a prescribed surface. The model also encompasses thermal effects in the bulk. The related PDE system for the displacements, the absolute temperature, and the delamination variable has a highly nonlinear character. On the contact surface, it features frictionless Signorini conditions and a nonconvex, brittle constraint acting as a transmission condition for the displacements. We prove the existence of (weak/energetic) solutions to the associated Cauchy problem, by approximating it in two steps with suitably regularized problems. We perform the two consecutive passages to the limit via refined variational convergence techniques.

R. Servadei, E. Valdinoci, The Brezis--Nirenberg result for the fractional Laplacian, Transactions of the American Mathematical Society, 367 (2015), pp. 67--102.

P. Colli, G. Gilardi, E. Rocca, J. Sprekels, Vanishing viscosities and error estimate for a Cahn--Hilliard type phase field system related to tumor growth, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 26 (2015), pp. 93--108.AbstractIn this paper we perform an asymptotic analysis for two different vanishing viscosity coefficients occurring in a phase field system of Cahn--Hilliard type that was recently introduced in order to approximate a tumor growth model. In particular, we extend some recent results obtained in [Colli-Gilardi-Hilhorst 2015], letting the two positive viscosity parameters tend to zero independently from each other and weakening the conditions on the initial data in such a way as to maintain the nonlinearities of the PDE system as general as possible. Finally, under proper growth conditions on the interaction potential, we prove an error estimate leading also to the uniqueness result for the limit system.

P. Colli, M.H. FarshbafShaker, G. Gilardi, J. Sprekels, Optimal boundary control of a viscous Cahn--Hilliard system with dynamic boundary condition and double obstacle potentials, SIAM Journal on Control and Optimization, 53 (2015), pp. 2696--2721.AbstractIn this paper, we investigate optimal boundary control problems for Cahn--Hilliard variational inequalities with a dynamic boundary condition involving double obstacle potentials and the Laplace--Beltrami operator. The cost functional is of standard tracking type, and box constraints for the controls are prescribed. We prove existence of optimal controls and derive first-order necessary conditions of optimality. The general strategy, which follows the lines of the recent approach by Colli, Farshbaf-Shaker, Sprekels (see Appl. Math. Optim., 2014) to the (simpler) Allen--Cahn case, is the following: we use the results that were recently established by Colli, Gilardi, Sprekels in the preprint arXiv:1407.3916 [math.AP] for the case of (differentiable) logarithmic potentials and perform a so-called “deep quench limit”. Using compactness and monotonicity arguments, it is shown that this strategy leads to the desired first-order necessary optimality conditions for the case of (non-differentiable) double obstacle potentials.

P. Colli, M.H. FarshbafShaker, G. Gilardi, J. Sprekels, Second-order analysis of a boundary control problem for the viscous Cahn--Hilliard equation with dynamic boundary conditions, Annals of the Academy of Romanian Scientists. Mathematics and its Applications., 7 (2015), pp. 41--66.AbstractIn this paper we establish second-order sufficient optimality conditions for a boundary control problem that has been introduced and studied by three of the authors in the preprint arXiv:1407.3916. This control problem regards the viscous Cahn--Hilliard equation with possibly singular potentials and dynamic boundary conditions.

P. Colli, M.H. FarshbafShaker, J. Sprekels, A deep quench approach to the optimal control of an Allen--Cahn equation with dynamic boundary conditions and double obstacles, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 71 (2015), pp. 1--24.AbstractIn this paper, we investigate optimal control problems for Allen-Cahn variational inequalities with a dynamic boundary condition involving double obstacle potentials and the Laplace-Beltrami operator. The approach covers both the cases of distributed controls and of boundary controls. The cost functional is of standard tracking type, and box constraints for the controls are prescribed. We prove existence of optimal controls and derive first-order necessary conditions of optimality. The general strategy is the following: we use the results that were recently established by two of the authors for the case of (differentiable) logarithmic potentials and perform a so-called “deep quench limit”. Using compactness and monotonicity arguments, it is shown that this strategy leads to the desired first-order necessary optimality conditions for the case of (non-differentiable) double obstacle potentials.

J. Dávila, M. DelPino, S. Dipierro, E. Valdinoci, Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum, Analysis & PDE, 8 (2015), pp. 1165--1235.AbstractFor a smooth, bounded Euclidean domain, we consider a nonlocal Schrödinger equation with zero Dirichlet datum. We construct a family of solutions that concentrate at an interior point of the domain in the form of a scaling of the ground state in entire space. Unlike the classical case, the leading order of the associated reduced energy functional in a variational reduction procedure is of polynomial instead of exponential order on the distance from the boundary, due to the nonlocal effect. Delicate analysis is needed to overcome the lack of localization, in particular establishing the rather unexpected asymptotics for the Green function in the expanding domain.

M. Egert, R. Haller-Dintelmann, J. Rehberg, Hardy's inequality for functions vanishing on a part of the boundary, Potential Analysis, 43 (2015), pp. 49--78.AbstractWe develop a geometric framework for Hardy's inequality on a bounded domain when the functions do vanish only on a closed portion of the boundary.

M.M. Fall, F. Mahmoudi, E. Valdinoci, Ground states and concentration phenomena for the fractional Schrödinger equation, Nonlinearity, 28 (2015), pp. 1937--1961.AbstractWe consider here solutions of the nonlinear fractional Schrödinger equation. We show that concentration points must be critical points for the potential. We also prove that, if the potential is coercive and has a unique global minimum, then ground states concentrate suitably at such minimal point. In addition, if the potential is radial, then the minimizer is unique.

E. Feireisl, E. Rocca, G. Schimperna, A. Zarnescu, Nonisothermal nematic liquid crystal flows with the Ball--Majumdar free energy, Annali di Matematica Pura ed Applicata. Serie Quarta. Fondazione Annali di Matematica Pura ed Applicata, c/o Dipartimento di Matematica ``U. Dini'', Firenze; Springer-Verlag, Heidelberg. English, French, German, Italian, English abstracts., 194 (2015), pp. 1269--1299.AbstractIn this paper we prove the existence of global in time weak solutions for an evolutionary PDE system modelling nonisothermal Landau-de Gennes nematic liquid crystal (LC) flows in three dimensions of space. In our model, the incompressible Navier-Stokes system for the macroscopic velocity $vu$ is coupled to a nonlinear convective parabolic equation describing the evolution of the Q-tensor $QQ$, namely a tensor-valued variable representing the normalized second order moments of the probability distribution function of the LC molecules. The effects of the (absolute) temperature $vt$ are prescribed in the form of an energy balance identity complemented with a global entropy production inequality. Compared to previous contributions, we can consider here the physically realistic singular configuration potential $f$ introduced by Ball and Majumdar. This potential gives rise to severe mathematical difficulties since it introduces, in the Q-tensor equation, a term which is at the same time singular in $QQ$ and degenerate in $vt$. To treat it a careful analysis of the properties of $f$, particularly of its blow-up rate, is carried out.

A. Fiscella, R. Servadei, E. Valdinoci, Asymptotically linear problems driven by fractional Laplacian operators, Mathematical Methods in the Applied Sciences, 38 (2015), pp. 3551--3563.AbstractIn this paper we study a non-local fractional Laplace equation, depending on a parameter, with asymptotically linear right-hand side. Our main result concerns the existence of weak solutions for this equation and it is obtained using variational and topological methods. We treat both the nonresonant case and the resonant one.

D.A. Gomes, S. Patrizi, Obstacle mean-field game problem, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 17 (2015), pp. 55--68.AbstractIn this paper, we introduce and study a first-order mean-field game obstacle problem. We examine the case of local dependence on the measure under assumptions that include both the logarithmic case and power-like nonlinearities. Since the obstacle operator is not differentiable, the equations for first-order mean field game problems have to be discussed carefully. Hence, we begin by considering a penalized problem. We prove this problem admits a unique solution satisfying uniform bounds. These bounds serve to pass to the limit in the penalized problem and to characterize the limiting equations. Finally, we prove uniqueness of solutions.

T. Roubíček, M. Thomas, Ch. Panagiotopoulos, Stress-driven local-solution approach to quasistatic brittle delamination, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 22 (2015), pp. 645--663.AbstractA unilateral contact problem between elastic bodies at small strains glued by a brittle adhesive is addressed in the quasistatic rate-independent setting. The delamination process is modelled as governed by stresses rather than by energies. This results in a specific scaling of an approximating elastic adhesive contact problem, discretised by a semi-implicit scheme and regularized by a BV-type gradient term. An analytical zero-dimensional example motivates the model and a specific local-solution concept. Two-dimensional numerical simulations performed on an engineering benchmark problem of debonding a fiber in an elastic matrix further illustrate the validity of the model, convergence, and algorithmical efficiency even for very rigid adhesives with high elastic moduli.

A.F.M. TERElst, J. Rehberg, Hölder estimates for second-order operators with mixed boundary conditions, Advances in Differential Equations, 20 (2015), pp. 299--360.AbstractIn this paper we investigate linear elliptic, second-order boundary value problems with mixed boundary conditions. Assuming only boundedness/ellipticity on the coefficient function and very mild conditions on the geometry of the domain -- including a very weak compatibility condition between the Dirichlet boundary part and its complement -- we prove first Hölder continuity of the solution. Secondly, Gaussian Hölder estimates for the corresponding heat kernel are derived. The essential instruments are De Giorgi and Morrey-Campanato estimates.

M. Thomas, Uniform Poincaré--Sobolev and relative isoperimetric inequalities for classes of domains, Discrete and Continuous Dynamical Systems, 35 (2015), pp. 2741--2761.AbstractThe aim of this paper is to prove an isoperimetric inequality relative to a d-dimensional, bounded, convex domain &Omega intersected with balls with a uniform relative isoperimetric constant, independent of the size of the radius r>0 and the position y∈cl(&Omega) of the center of the ball. For this, uniform Sobolev, Poincaré and Poincaré-Sobolev inequalities are deduced for classes of (not necessarily convex) domains that satisfy a uniform cone property. It is shown that the constants in all of these inequalities solely depend on the dimensions of the cone, space dimension d, the diameter of the domain and the integrability exponent p∈[1,d).

N. Ahmed, G. Matthies, Higher order continuous Galerkin--Petrov time stepping schemes for transient convection-diffusion-reaction equations, ESAIM: Mathematical Modelling and Numerical Analysis, 49 (2015), pp. 1429--1450.AbstractWe present the analysis for the higher order continuous Galerkin--Petrov (cGP) time discretization schemes in combination with the one-level local projection stabilization in space applied to time-dependent convection-diffusion-reaction problems. Optimal a-priori error estimates will be proved. Numerical studies support the theoretical results. Furthermore, a numerical comparison between continuous Galerkin--Petrov and discontinuous Galerkin time discretization schemes will be given.

W. Dreyer, R. Huth, A. Mielke, J. Rehberg, M. Winkler, Global existence for a nonlocal and nonlinear Fokker--Planck equation, ZAMP Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 66 (2015), pp. 293--315.AbstractWe consider a Fokker-Planck equation on a compact interval where, as a constraint, the first moment is a prescribed function of time. Eliminating the associated Lagrange multiplier one obtains nonlinear and nonlocal terms. After establishing suitable local existence results, we use the relative entropy as an energy functional. However, the time-dependent constraint leads to a source term such that a delicate analysis is needed to show that the dissipation terms are strong enough to control the work done by the constraint. We obtain global existence of solutions as long as the prescribed first moment stays in the interior of an interval. If the prescribed moment converges to a constant value inside the interior of the interval, then the solution stabilises to the unique steady state.

M.H. FarshbafShaker, Ch. Heinemann, A phase field approach for optimal boundary control of damage processes in two-dimensional viscoelastic media, Mathematical Models & Methods in Applied Sciences, 25 (2015), pp. 2749--2793.AbstractIn this work we investigate a phase field model for damage processes in two-dimensional viscoelastic media with nonhomogeneous Neumann data describing external boundary forces. In the first part we establish global-in-time existence, uniqueness, a priori estimates and continuous dependence of strong solutions on the data. The main difficulty is caused by the irreversibility as well as boundedness of the phase field variable which results in a doubly constrained PDE system. In the last part we consider an optimal control problem where a cost functional penalizes maximal deviations from prescribed damage profiles. The goal is to minimize the cost functional with respect to exterior forces acting on the boundary which play the role of the control variable in the considered model . To this end, we prove existence of minimizers and study a family of “local” approximations via adapted cost functionals.

H. Hanke, D. Knees, Homogenization of elliptic systems with non-periodic, state dependent coefficients, Asymptotic Analysis, 92 (2015), pp. 203--234.AbstractIn this paper, a homogenization problem for an elliptic system with non-periodic, state dependent coefficients representing microstructure is investigated. The state functions defining the tensor of coefficients are assumed to have an intrinsic length scale denoted by ε > 0. The aim is the derivation of an effective model by investigating the limit process ε → 0 of the state functions rigorously. The effective model is independent of the parameter ε > 0 but preserves the microscopic structure of the state functions (ε > 0), meaning that the effective tensor is given by a unit cell problem prescribed by a suitable microscopic tensor. Due to the non-periodic structure of the state functions and the corresponding microstructure, the effective tensor turns out to vary from point to point (in contrast to a periodic microscopic model). In a forthcoming paper, these states will be solutions of an additional evolution law describing changes of the microstructure. Such changes could be the consequences of temperature changes, phase separation or damage progression, for instance. Here, in addition to the above and as a preparation for an application to time-dependent damage models (discussed in a future paper), we provide a Γ-convergence result of sequences of functionals being related to the previous microscopic models with state dependent coefficients. This requires a penalization term for piecewise constant state functions that allows us to extract from bounded sequences those sequences converging to a Sobolev function in some sense. The construction of the penalization term is inspired by techniques for Discontinuous Galerkin methods and is of own interest. A compactness and a density result are provided.

CH. Heinemann, Ch. Kraus, A degenerating Cahn--Hilliard system coupled with complete damage processes, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 22 (2015), pp. 388--403.AbstractComplete damage in elastic solids appears when the material looses all its integrity due to high exposure. In the case of alloys, the situation is quite involved since spinodal decomposition and coarsening also occur at sufficiently low temperatures which may lead locally to high stress peaks. Experimental observations on solder alloys reveal void and crack growth especially at phase boundaries. In this work, we investigate analytically a degenerating PDE system with a time-depending domain for phase separation and complete damage processes under time-varying Dirichlet boundary conditions. The evolution of the system is described by a degenerating parabolic differential equation of fourth order for the concentration, a doubly nonlinear differential inclusion for the damage process and a degenerating quasi-static balance equation for the displacement field. All these equations are strongly nonlinearly coupled. Because of the doubly degenerating character and the doubly nonlinear differential inclusion we are confronted with introducing a suitable notion of weak solutions. We choose a notion of weak solutions which consists of weak formulations of the diffusion equation and the momentum balance, a one-sided variational inequality for the damage function and an energy estimate. For the introduced degenerating system, we prove existence of weak solutions in an $SBV$-framework. The existence result is based on an approximation system, where the elliptic degeneracy of the displacement field and the parabolic degeneracy of the concentration are eliminated. In the framework of phase separation and damage, this means that the approximation system allows only for partial damage and a non-degenerating mobility tensor. For the approximation system, existence results are established. Then, a passage to the limit shows existence of weak solutions of the degenerating system.

CH. Heinemann, Ch. Kraus, Complete damage in linear elastic materials -- Modeling, weak formulation and existence results, Calculus of Variations and Partial Differential Equations, 54 (2015), pp. 217--250.AbstractWe introduce a complete damage model with a time-depending domain for linear-elastically stressed solids under time-varying Dirichlet boundary conditions. The evolution of the system is described by a doubly nonlinear differential inclusion for the damage process and a quasi-static balance equation for the displacement field. For the introduced complete damage model, we propose a classical formulation and a corresponding suitable weak formulation in an $SBV$-framework. We show that the classical differential inclusion can be regained from the notion of weak solutions under certain regularity assumptions. The main aim of this work is to prove local-in-time existence and global-in-time existence in some weaker sense for the introduced model. In contrast to incomplete damage theories, the material can be exposed to damage in the proposed model in such a way that the elastic behavior may break down on the damaged parts of the material, i.e. we loose coercivity properties of the free energy. This leads to several mathematical difficulties. For instance, it might occur that not completely damaged material regions are isolated from the Dirichlet boundary. In this case, the deformation field cannot be controlled in the transition from incomplete to complete damage. To tackle this problem, we consider the evolution process on a time-depending domain. In this context, two major challenges arise: Firstly, the time-dependent domain approach leads to jumps in the energy which have to be accounted for in the energy inequality of the notion of weak solutions. To handle this problem, several energy estimates are established by $Gamma$-convergence techniques. Secondly, the time-depending domain might have bad smoothness properties such that Korn's inequality cannot be applied. To this end, a covering result for such sets with smooth compactly embedded domains has been shown.

CH. Heinemann, Ch. Kraus, Existence of weak solutions for a PDE system describing phase separation and damage processes including inertial effects, Discrete and Continuous Dynamical Systems, 35 (2015), pp. 2565--2590.AbstractIn this paper, we consider a coupled PDE system describing phase separation and damage phenomena in elastically stressed alloys in the presence of inertial effects. The material is considered on a bounded Lipschitz domain with mixed boundary conditions for the displacement variable. The main aim of this work is to establish existence of weak solutions for the introduced hyperbolic-parabolic system. To this end, we first generalize the notion of weak solution introduced in WIAS 1520. Then we prove existence of weak solutions by means of regularization, time-discretization and different variational techniques.

CH. Heinemann, Ch. Kraus, Existence of weak solutions for a hyperbolic-parabolic phase field system with mixed boundary conditions on non-smooth domains, SIAM Journal on Mathematical Analysis, 47 (2015), pp. 2044--2073.AbstractThe aim of this paper is to prove existence of weak solutions of hyperbolic-parabolic evolution inclusions defined on Lipschitz domains with mixed boundary conditions describing, for instance, damage processes and elasticity with inertial effects. To this end, we first present a suitable weak formulation in order to deal with such evolution inclusions. Then, existence of weak solutions is proven by utilizing time-discretization, $H^2$--regularization and variational techniques.

CH. Heinemann, E. Rocca, Damage processes in thermoviscoelastic materials with damage-dependent thermal expansion coefficients, Mathematical Methods in the Applied Sciences, 38 (2015), pp. 4587--4612.AbstractIn this paper we prove existence of global in time weak solutions for a highly nonlinear PDE system arising in the context of damage phenomena in thermoviscoelastic materials. The main novelty of the present contribution with respect to the ones already present in the literature consists in the possibility of taking into account a damage-dependent thermal expansion coefficient. This term implies the presence of nonlinear couplings in the PDE system, which make the analysis more challenging.

A. Mielke, J. Naumann, Global-in-time existence of weak solutions to Kolmogorov's two-equation model of turbulence, Comptes Rendus Mathematique. Academie des Sciences. Paris, 353 (2015), pp. 321--326.AbstractWe consider Kolmogorov's model for the turbulent motion of an incompressible fluid in ℝ3. This model consists in a Navier-Stokes type system for the mean flow u and two further partial differential equations: an equation for the frequency ω and for the kinetic energy k each. We investigate this system of partial differential equations in a cylinder Ω x ]0,T[ (Ω ⊂ ℝ3 cube, 0 < T < +∞) under spatial periodic boundary conditions on ∂Ω x ]0,T[ and initial conditions in Ω x {0}. We present an existence result for a weak solution {u, ω, k} to the problem under consideration, with ω, k obeying the inequalities and .

A. Mielke, Deriving amplitude equations via evolutionary Gamma convergence, Discrete and Continuous Dynamical Systems, 35 (2015), pp. 2679--2700.AbstractWe discuss the justification of the Ginzburg-Landau equation with real coefficients as an amplitude equation for the weakly unstable one-dimensional Swift-Hohenberg equation. In contrast to classical justification approaches we employ the method of evolutionary Gamma convergence by reformulating both equations as gradient systems. Using a suitable linear transformation we show Gamma convergence of the associated energies in suitable function spaces. The limit passage of the time-dependent problem relies on the recent theory of evolutionary variational inequalities for families of uniformly convex functionals as developed by Daneri and Savaré 2010. In the case of a cubic energy it suffices that the initial conditions converge strongly in L2, while for the case of a quadratic nonlinearity we need to impose weak convergence in H1. However, we do not need wellpreparedness of the initial conditions.

A. Mielke, J. Haskovec, P.A. Markowich, On uniform decay of the entropy for reaction-diffusion systems, Journal of Dynamics and Differential Equations, 27 (2015), pp. 897--928.AbstractIn this work we derive entropy decay estimates for a class of nonlinear reaction-diffusion systems modeling reversible chemical reactions under the assumption of detailed balance. In particular, we provide explicit bounds for the exponential decay of the relative logarithmic entropy, being based essentially on the application of the log-Sobolev inequality and a convexification argument only, making it quite robust to model variations. An important feature of our analysis is the interaction of the two different dissipative mechanisms: pure diffusion, forcing the system asymptotically to the homogeneous state, and pure reaction, forcing the solution to the (possibly inhomogeneous) chemical equilibrium. Only the interaction of both mechanisms provides the convergence to the homogeneous equilibrium. Moreover, we introduce two generalizations of the main result: we allow for vanishing diffusion constants in some chemical components, and we consider different entropy functionals. We provide a few examples to highlight the usability of our approach and shortly discuss possible further applications and open questions.

S. Heinz, On the structure of the quasiconvex hull in planar elasticity, Calculus of Variations and Partial Differential Equations, 50 (2014), pp. 481--489.AbstractLet K and L be compact sets of real 2x2 matrices with positive determinant. Suppose that both sets are frame invariant, meaning invariant under the left action of the special orthogonal group. Then we give an algebraic characterization for K and L to be incompatible for homogeneous gradient Young measures. This result permits a simplified characterization of the quasiconvex hull and the rank-one convex hull in planar elasticity.

P. Colli, G. Gilardi, P. Krejčí, P. Podio-Guidugli, J. Sprekels, Analysis of a time discretization scheme for a nonstandard viscous Cahn--Hilliard system, ESAIM: Mathematical Modelling and Numerical Analysis, 48 (2014), pp. 1061--1087.AbstractIn this paper we propose a time discretization of a system of two parabolic equations describing diffusion-driven atom rearrangement in crystalline matter. The equations express the balances of microforces and microenergy; the two phase fields are the order parameter and the chemical potential. The initial and boundary-value problem for the evolutionary system is known to be well posed. Convergence of the discrete scheme to the solution of the continuous problem is proved by a careful development of uniform estimates, by weak compactness and a suitable treatment of nonlinearities. Moreover, for the difference of discrete and continuous solutions we prove an error estimate of order one with respect to the time step.

P. Colli, G. Gilardi, P. Krejčí, J. Sprekels, A continuous dependence result for a nonstandard system of phase field equations, Mathematical Methods in the Applied Sciences, 37 (2014), pp. 1318--1324.AbstractThe present note deals with a nonstandard systems of differential equations describing a two-species phase segregation. This system naturally arises in the asymptotic analysis carried out recently by the same authors, as the diffusion coefficient in the equation governing the evolution of the order parameter tends to zero. In particular, an existence result has been proved for the limit system in a very general framework. On the contrary, uniqueness was shown by assuming a constant mobility coefficient. Here, we generalize this result and prove a continuous dependence property in the case that the mobility coefficient suitably depends on the chemical potential.

P. Colli, G. Gilardi, P. Krejčí, J. Sprekels, A vanishing diffusion limit in a nonstandard system of phase field equations, Evolution Equations and Control Theory, 3 (2014), pp. 257--275.AbstractWe are concerned with a nonstandard phase field model of Cahn--Hilliard type. The model, which was introduced by Podio-Guidugli (Ric. Mat. 2006), describes two-species phase segregation and consists of a system of two highly nonlinearly coupled PDEs. It has been recently investigated by Colli, Gilardi, Podio-Guidugli, and Sprekels in a series of papers: see, in particular, SIAM J. Appl. Math. 2011, and Boll. Unione Mat. Ital. 2012. In the latter contribution, the authors can treat the very general case in which the diffusivity coefficient of the parabolic PDE is allowed to depend nonlinearly on both variables. In the same framework, this paper investigates the asymptotic limit of the solutions to the initial-boundary value problems as the diffusion coefficient $sigma$ in the equation governing the evolution of the order parameter tends to zero. We prove that such a limit actually exists and solves the limit problem, which couples a nonlinear PDE of parabolic type with an ODE accounting for the phase dynamics. In the case of a constant diffusivity, we are able to show uniqueness and to improve the regularity of the solution.

P. Colli, G. Gilardi, J. Sprekels, On the Cahn--Hilliard equation with dynamic boundary conditions and a dominating boundary potential, Journal of Mathematical Analysis and Applications, 419 (2014), pp. 972--994.AbstractThe Cahn--Hilliard and viscous Cahn--Hilliard equations with singular and possibly nonsmooth potentials and dynamic boundary condition are considered and some well-posedness and regularity results are proved.

A. Gloria, S. Neukamm, F. Otto, An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations, ESAIM: Mathematical Modelling and Numerical Analysis, 48 (2014), pp. 325--346.AbstractWe establish an optimal, linear rate of convergence for the stochastic homogenization of discrete linear elliptic equations. We consider the model problem of independent and identically distributed coefficients on a discretized unit torus. We show that the difference between the solution to the random problem on the discretized torus and the first two terms of the two-scale asymptotic expansion has the same scaling as in the periodic case. In particular the L2-norm in probability of the H1-norm in space of this error scales like ε, where ε is the discretization parameter of the unit torus. The proof makes extensive use of previous results by the authors, and of recent annealed estimates on the Greens function by Marahrens and the third author.

R. Haller-Dintelmann, W. Höppner, H.-Chr. Kaiser, J. Rehberg, G. Ziegler, Optimal elliptic Sobolev regularity near three-dimensional, multi-material Neumann vertices, Functional Analysis and its Applications, 48 (2014), pp. 208--222.AbstractWe investigate optimal elliptic regularity (within the scale of Sobolev spaces) of anisotropic div-grad operators in three dimensions at a multi-material vertex on the Neumann boundary part of the polyhedral spatial domain. The gradient of a solution to the corresponding elliptic PDE (in a neighbourhood of the vertex) is integrable to an index greater than three.

S. Melchionna, E. Rocca, On a nonlocal Cahn--Hilliard equation with a reaction term, Advances in Mathematical Sciences and Applications, 24 (2014), pp. 461--497.AbstractWe prove existence, uniqueness, regularity and separation properties for a nonlocal Cahn- Hilliard equation with a reaction term. We deal here with the case of logarithmic potential and degenerate mobility as well an uniformly lipschitz in u reaction term g(x, t, u).

A. Mielke, S. Reichelt, M. Thomas, Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion, Networks Heterogeneous Media, 9 (2014), pp. 353--382.AbstractWe derive a two-scale homogenization limit for reaction-diffusion systems where for some species the diffusion length is of order 1 whereas for the other species the diffusion length is of the order of the periodic microstructure. Thus, in the limit the latter species will display diffusion only on the microscale but not on the macroscale. Because of this missing compactness, the nonlinear coupling through the reaction terms cannot be homogenized but needs to be treated on the two-scale level. In particular, we have to develop new error estimates to derive strong convergence results for passing to the limit.

A. Mielke, Ch. Ortner, Y. Şengül, An approach to nonlinear viscoelasticity via metric gradient flows, SIAM Journal on Mathematical Analysis, 46 (2014), pp. 1317--1347.AbstractWe formulate quasistatic nonlinear finite-strain viscoelasticity of rate-type as a gradient system. Our focus is on nonlinear dissipation functionals and distances that are related to metrics on weak diffeomorphisms and that ensure time-dependent frame-indifference of the viscoelastic stress. In the multidimensional case we discuss which dissipation distances allow for the solution of the time-incremental problem. Because of the missing compactness the limit of vanishing timesteps can only be obtained by proving some kind of strong convergence. We show that this is possible in the one-dimensional case by using a suitably generalized convexity in the sense of geodesic convexity of gradient flows. For a general class of distances we derive discrete evolutionary variational inequalities and are able to pass to the time-continuous in some case in a specific case.

A. Mielke, M.A. Peletier, D.R.M. Renger, On the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion, Potential Analysis, 41 (2014), pp. 1293--1325.AbstractMotivated by the occurence in rate functions of time-dependent large-deviation principles, we study a class of non-negative functions ℒ that induce a flow, given by ℒ(zt,żt)=0. We derive necessary and sufficient conditions for the unique existence of a generalized gradient structure for the induced flow, as well as explicit formulas for the corresponding driving entropy and dissipation functional. In particular, we show how these conditions can be given a probabilistic interpretation when ℒ is associated to the large deviations of a microscopic particle system. Finally, we illustrate the theory for independent Brownian particles with drift, which leads to the entropy-Wasserstein gradient structure, and for independent Markovian particles on a finite state space, which leads to a previously unknown gradient structure.

A. Mielke, S. Zelik, On the vanishing-viscosity limit in parabolic systems with rate-independent dissipation terms, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V, XIII (2014), pp. 67--135.AbstractWe consider quasilinear parabolic systems with a nonsmooth rate-independent dissipation term in the limit of very slow loading rates, or equivalently with fixed loading and vanishing viscosity $varepsilon>0$. Because for nonconvex energies the solutions will develop jumps, we consider the vanishing-viscosity limit for the graphs of the solutions in the extended state space in arclength parametrization, where the norm associated with the viscosity is used to keep the subdifferential structure of the problem. A crucial point in the analysis are new a priori estimates that are rate independent and that allows us to show that the total length of the graph remains bounded in the vanishing-viscosity limit. To derive these estimates we combine parabolic regularity estimates with ideas from rate-independent systems.

S. Neukamm, H. Olbermann, Homogenization of the nonlinear bending theory for plates, Calculus of Variations and Partial Differential Equations, (published online on Sept. 14, 2014), DOI 10.1007/s00526-014-0765-2 .AbstractWe carry out the spatially periodic homogenization of nonlinear bending theory for plates. The derivation is rigorous in the sense of Gamma-convergence. In contrast to what one naturally would expect, our result shows that the limiting functional is not simply a quadratic functional of the second fundamental form of the deformed plate as it is the case in nonlinear plate theory. It turns out that the limiting functional discriminates between whether the deformed plate is locally shaped like a "cylinder" or not. For the derivation we investigate the oscillatory behavior of sequences of second fundamental forms associated with isometric immersions, using two-scale convergence. This is a non-trivial task, since one has to treat two-scale convergence in connection with a nonlinear differential constraint.

P.-É. Druet, Global Lipschitz continuity for elliptic transmission problems with a boundary intersecting interface, Mathematica Bohemica, 138 (2013), pp. 185--224.AbstractWe investigate the regularity of the weak solution to elliptic transmission problems that involve two layered anisotropic materials separated by a boundary intersecting interface. Under a compatibility condition for the angle of contact of the two surfaces and the boundary data, we prove the existence of square-integrable second derivatives, and the global Lipschitz continuity of the solution. We show that the second weak derivatives remain integrable to a certain power less than two if the compatibility condition is violated.

CH. Heinemann, Ch. Kraus, Existence results for diffuse interface models describing phase separation and damage, European Journal of Applied Mathematics, 24 (2013), pp. 179--211.AbstractIn this paper we analytically investigate Cahn-Hilliard and Allen-Cahn systems which are coupled with elasticity and uni-directional damage processes. We are interested in the case where the free energy contains logarithmic terms of the chemical concentration variable and quadratic terms of the gradient of the damage variable. For elastic Cahn-Hilliard and Allen-Cahn systems coupled with uni-directional damage processes, an appropriate notion of weak solutions is presented as well as an existence result based on certain regularization methods and an higher integrability result for the strain.

K. Götze, Strong solutions for the interaction of a rigid body and a viscoelastic fluid, Journal of Mathematical Fluid Mechanics, 15 (2013), pp. 663--688.AbstractWe study a coupled system of equations describing the movement of a rigid body which is immersed in a viscoelastic fluid. It is shown that under natural assumptions on the data and for general goemetries of the rigid body, excluding contact scenarios, a unique local-in-time strong solution exists.

M. Liero, U. Stefanelli, A new minimum principle for Lagrangian mechanics, Journal of Nonlinear Science, 23 (2013), pp. 179--204.AbstractWe present a novel variational approach to Lagrangian mechanics based on elliptic regularization with respect to time. A class of parameter-dependent global-in-time minimization problems is presented and the convergence of the respective minimizers to the solution of the system of Lagrange's equations is ascertained. Moreover, we extend this perspective to mixed dissipative/nondissipative situations, present a finite time-horizon version of this approach, and provide related Gamma-convergence results. Finally, some discussion on corresponding time-discrete versions of the principle is presented.

M. Liero, A. Mielke, Gradient structures and geodesic convexity for reaction-diffusion systems, Philosophical Transactions of the Royal Society A : Mathematical, Physical & Engineering Sciences, 371 (2013), pp. 20120346/1--20120346/28.AbstractWe consider systems of reaction-diffusion equations as gradient systems with respect to an entropy functional and a dissipation metric given in terms of a so-called Onsager operator, which is a sum of a diffusion part of Wasserstein type and a reaction part. We provide methods for establishing geodesic lambda-convexity of the entropy functional by purely differential methods, thus circumventing arguments from mass transportation. Finally, several examples, including a drift-diffusion system, provide a survey on the applicability of the theory. We consider systems of reaction-diffusion equations as gradient systems with respect to an entropy functional and a dissipation metric given in terms of a so-called Onsager operator, which is a sum of a diffusion part of Wasserstein type and a reaction part. We provide methods for establishing geodesic lambda-convexity of the entropy functional by purely differential methods, thus circumventing arguments from mass transportation. Finally, several examples, including a drift-diffusion system, provide a survey on the applicability of the theory.

M. Liero, Passing from bulk to bulk/surface evolution in the Allen--Cahn equation, NoDEA. Nonlinear Differential Equations and Applications, 20 (2013), pp. 919--942.AbstractIn this paper we formulate a boundary layer approximation for an Allen-Cahn-type equation involving a small parameter $eps$. Here, $eps$ is related to the thickness of the boundary layer and we are interested in the limit when $eps$ tends to 0 in order to derive nontrivial boundary conditions. The evolution of the system is written as an energy balance formulation of the L^2-gradient flow with the corresponding Allen-Cahn energy functional. By transforming the boundary layer to a fixed domain we show the convergence of the solutions to a solution of a limit system. This is done by using concepts related to Gamma- and Mosco convergence. By considering different scalings in the boundary layer we obtain different boundary conditions.

S. Neukamm, I. Velcic, Derivation of a homogenized von-Kármán plate theory from 3D nonlinear elasticity, Mathematical Models & Methods in Applied Sciences, 23 (2013), pp. 2701--2748.AbstractWe rigorously derive a homogenized von-Kármán plate theory as a Gamma-limit from nonlinear three-dimensional elasticity by combining homogenization and dimension reduction. Our starting point is an energy functional that describes a nonlinear elastic, three-dimensional plate with spatially periodic material properties. The functional features two small length scales: the period $epsilon$ of the elastic composite material, and the thickness h of the slender plate. We study the behavior as $epsilon$ and h simultaneously converge to zero in the von-Kármán scaling regime. The obtained limit is a homogenized von-Kármán plate model. Its effective material properties are determined by a relaxation formula that exposes a non-trivial coupling of the behavior of the out-of-plane displacement with the oscillatory behavior in the in-plane directions. In particular, the homogenized coefficients depend on the relative scaling between h and $epsilon$, and different values arise for h<

A. Fiaschi, D. Knees, S. Reichelt, Global higher integrability of minimizers of variational problems with mixed boundary conditions, Journal of Mathematical Analysis and Applications, 401 (2013), pp. 269--288.AbstractWe consider integral functionals with densities of p-growth, with respect to gradients, on a Lipschitz domain with mixed boundary conditions. The aim of this paper is to prove that, under uniform estimates within certain classes of p-growth and coercivity assumptions on the density, the minimizers are of higher integrability order, meaning that they belong to the space of first order Sobolev functions with an integrability of order p+ε for a uniform ε >0. The results are applied to a model describing damage evolution in a nonlinear elastic body and to a model for shape memory alloys.

P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, An asymptotic analysis for a nonstandard Cahn--Hilliard system with viscosity, Discrete and Continuous Dynamical Systems -- Series S, 6 (2013), pp. 353--368.AbstractThis paper is concerned with a diffusion model of phase-field type, consisting of a parabolic system of two partial differential equations, interpreted as balances of microforces and microenergy, for two unknowns: the problem's order parameter $rho$ and the chemical potential $mu$; each equation includes a viscosity term -- respectively, $varepsilon,partial_tmu$ and $delta,partial_trho$ -- with $varepsilon$ and $delta$ two positive parameters; the field equations are complemented by Neumann homogeneous boundary conditions and suitable initial conditions. In a recent paper [5], we proved that this problem is well-posed and investigated the long-time behavior of its $(varepsilon,delta)-$solutions. Here we discuss the asymptotic limit of the system as $eps$ tends to 0. We prove convergence of $(varepsilon,delta)-$solutions to the corresponding solutions for the case $eps$ =0, whose long-time behavior we characterize; in the proofs, we employ compactness and monotonicity arguments.

P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, Global existence and uniqueness for a singular/degenerate Cahn--Hilliard system with viscosity, Journal of Differential Equations, 254 (2013), pp. 4217--4244.AbstractExistence and uniqueness are investigated for a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. This system aims to model two-species phase segregation on an atomic [19]; in the balance equations of microforces and microenergy, the two unknowns are the order parameter $rho$ and the chemical potential $mu$. A simpler version of the same system has recently been discussed in [8]. In this paper, a fairly more general phase-field equation for $rho$ is coupled with a genuinely nonlinear diffusion equation for $mu$. The existence of a global-in-time solution is proved with the help of suitable a priori estimates. In the case of costant atom mobility, a new and rather unusual uniqueness proof is given, based on a suitable combination of variables.

M. Geissert, K. Götze, M. Hieber, $L^p$-theory for strong solutions to fluid-rigid body interaction in Newtonian and generalized Newtonian fluids, Transactions of the American Mathematical Society, 365 (2013), pp. 1393--1439.AbstractConsider the system of equations describing the motion of a rigid body immersed in a viscous, incompressible fluid of Newtonian or generalized Newtonian type. The class of fluids considered includes in particular shear-thinning or shear-thickening fluids of power-law type of exponent $ dgeq 1$. We develop a method to prove that this system admits a unique, local, strong solution in the $ L^p$-setting. The approach presented in the case of generalized Newtonian fluids is based on the theory of quasi-linear evolution equations and requires that the exponent $ p$ satisfies the condition $ p>5$.

A. Glitzky, A. Mielke, A gradient structure for systems coupling reaction-diffusion effects in bulk and interfaces, ZAMP Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 64 (2013), pp. 29--52.AbstractWe derive gradient-flow formulations for systems describing drift-diffusion processes of a finite number of species which undergo mass-action type reversible reactions. Our investigations cover heterostructures, where material parameter may depend in a nonsmooth way on the space variable. The main results concern a gradient flow formulation for electro-reaction-diffusion systems with active interfaces permitting drift-diffusion processes and reactions of species living on the interface and transfer mechanisms allowing bulk species to jump into an interface or to pass through interfaces. The gradient flows are formulated in terms of two functionals: the free energy and the dissipation potential. Both functionals consist of a bulk and an interface integral. The interface integrals determine the interface dynamics as well as the self-consistent coupling to the model in the bulk. The advantage of the gradient structure is that it automatically generates thermodynamically consistent models.

D. Knees, R. Rossi, Ch. Zanini, A vanishing viscosity approach to a rate-independent damage model, Mathematical Models & Methods in Applied Sciences, 23 (2013), pp. 565--616.AbstractWe analyze a rate-independent model for damage evolution in elastic bodies. The central quantities are a stored energy functional and a dissipation functional, which is assumed to be positively homogeneous of degree one. Since the energy is not simultaneously (strictly) convex in the damage variable and the displacements, solutions may have jumps as a function of time. The latter circumstance makes it necessary to recur to suitable notions of weak solution. However, the by-now classical concept of global energetic solution fails to describe accurately the behavior of the system at jumps. Hence, we consider rate-independent damage models as limits of systems driven by viscous, rate-dependent dissipation. We use a technique for taking the vanishing viscosity limit, which is based on arc-length reparameterization. In this way, in the limit we obtain a novel formulation for the rate-independent damage model, which highlights the interplay of viscous and rate-independent effects in the jump regime, and provides a better description of the energetic behavior of the system at jumps.

D. Knees, A. Schröder, Computational aspects of quasi-static crack propagation, Discrete and Continuous Dynamical Systems -- Series S, 6 (2013), pp. 63--99.AbstractThe focus of this note lies on the numerical analysis of models describing the propagation of a single crack in a linearly elastic material. The evolution of the crack is modeled as a rate-independent process based on the Griffith criterion. We follow two different approaches for setting up mathematically well defined models: the global energetic approach and an approach based on a viscous regularization. We prove the convergence of solutions of fully discretized models (i.e. with respect to time and space) and derive relations between the discretization parameters (mesh size, time step size, viscosity parameter, crack increment) which guarantee the convergence of the schemes. Further, convergence rates are provided for the approximation of energy release rates by certain discrete energy release rates. Thereby we discuss both, models with self-contact conditions on the crack faces as well as models with pure Neumann conditions on the crack faces. The convergence proofs rely on regularity estimates for the elastic fields close to the crack tip and local and global finite element error estimates. Finally the theoretical results are illustrated with some numerical calculations.

M. Thomas, Quasistatic damage evolution with spatial BV-regularization, Discrete and Continuous Dynamical Systems -- Series S, 6 (2013), pp. 235--255.AbstractAn existence result for energetic solutions of rate-independent damage processes is established. We consider a body consisting of a physically linearly elastic material undergoing infinitesimally small deformations and partial damage. In [ThomasMielke10DamageZAMM] an existence result in the small strain setting was obtained under the assumption that the damage variable z satisfies z∈ W1,r(Ω) with r∈(1,∞) for Ω⊂Rd. We now cover the case r=1. The lack of compactness in W1,1(Ω) requires to do the analysis in BV(Ω). This setting allows it to consider damage variables with values in 0,1. We show that such a brittle damage model is obtained as the Γ-limit of functionals of Modica-Mortola type.

D. Hömberg, K. Krumbiegel, J. Rehberg, Optimal control of a parabolic equation with dynamic boundary condition, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 67 (2013), pp. 3--31.AbstractWe investigate a control problem for the heat equation. The goal is to find an optimal heat transfer coefficient in the Robin boundary condition such that a desired temperature distribution at the boundary is adhered. To this end we consider a function space setting in which the heat flux across the boundary is forced to be an $L^p$ function with respect to the surface measure, which in turn implies higher regularity for the time derivative of temperature. We show that the corresponding elliptic operator generates a strongly continuous semigroup of contractions and apply the concept of maximal parabolic regularity. This allows to show the existence of an optimal control and the derivation of necessary and sufficient optimality conditions.

A. Mielke, R. Rossi, G. Savaré, Nonsmooth analysis of doubly nonlinear evolution equations, Calculus of Variations and Partial Differential Equations, 46 (2013), pp. 253--310.AbstractIn this paper we analyze a broad class of abstract doubly nonlinear evolution equations in Banach spaces, driven by nonsmooth and nonconvex energies. We provide some general sufficient conditions, on the dissipation potential and the energy functional, for existence of solutions to the related Cauchy problem. We prove our main existence result by passing to the limit in a time-discretization scheme with variational techniques. Finally, we discuss an application to a material model in finite-strain elasticity.

A. Mielke, E. Rohan, Homogenization of elastic waves in fluid-saturated porous media using the Biot model, Mathematical Models & Methods in Applied Sciences, 23 (2013), pp. 873--916.AbstractWe consider periodically heterogeneous fluid-saturated poroelastic media described by the Biot model with inertia effects. The weak and semistrong formulations for displacement, seepage and pressure fields involve three equations expressing the momentum and mass balance and the Darcy law. Using the two-scale homogenization method we obtain the limit two-scale problem and prove the existence and uniqueness of its weak solutions. The Laplace transformation in time is used to decouple the macroscopic and microscopic scales. It is shown that the seepage velocity is eliminated form the macroscopic equations involving strain and pressure fields only. The plane harmonic wave propagation is studied using an example of layered medium. Illustrations show some influence of the orthotropy on the dispersion phenomena.

A. Mielke, Geodesic convexity of the relative entropy in reversible Markov chains, Calculus of Variations and Partial Differential Equations, 48 (2013), pp. 1-31.AbstractWe consider finite-dimensional, time-continuous Markov chains satisfying the detailed balance condition as gradient systems with the relative entropy E as driving functional. The Riemannian metric is defined via its inverse matrix called the Onsager matrix K. We provide methods for establishing geodesic λ-convexity of the entropy and treat several examples including some more general nonlinear reaction systems

A. Mielke, Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions, Discrete and Continuous Dynamical Systems -- Series S, 6 (2013), pp. 479--499.AbstractWe show that many couplings between parabolic systems for processes in solids can be formulated as a gradient system with respect to the total free energy or the total entropy. This includes Allen-Cahn, Cahn-Hilliard, and reaction-diffusion systems and the heat equation. For this, we write the coupled system as an Onsager system (X,Φ,K) defining the evolution $dot U$= - K(U) DΦ(U). Here Φ is the driving functional, while the Onsager operator K(U) is symmetric and positive semidefinite. If the inverse G=K-1 exists, the triple (X,Φ,G) defines a gradient system. Onsager systems are well suited to model bulk-interface interactions by using the dual dissipation potential Ψ*(U, Ξ)= ½ 〈Ξ K(U) Ξ〉. Then, the two functionals Φ and Ψ* can be written as a sum of a volume integral and a surface integral, respectively. The latter may contain interactions of the driving forces in the interface as well as the traces of the driving forces from the bulk. Thus, capture and escape mechanisms like thermionic emission appear naturally in Onsager systems, namely simply through integration by parts.

P.-E. Druet, The classical solvability of the contact angle problem for generalized equations of mean curvature type, Portugaliae Mathematica. A Journal of the Portuguese Mathematical Society, 69 (2012), pp. 233--258.AbstractIn this paper, mean curvature type equations with general potentials and contact angle boundary conditions are considered. We extend the ideas of Ural'tseva, formulating sharper hypotheses for the existence of a classical solution. Corner stone for these results is a method to estimate quantities on the boundary of the free surface. We moreover provide alternative proofs for the higher-order estimates, and for the existence result.

M. Liero, Th. Roche, Rigorous derivation of a plate theory in linear elastoplasticity via $Gamma$-convergence, NoDEA. Nonlinear Differential Equations and Applications, 19 (2012), pp. 437--457.AbstractThis paper deals with dimension reduction in linearized elastoplasticity in the rate-independent case. The reference configuration of the elastoplastic body is given by a two-dimensional middle surface and a small but positive thickness. We derive a limiting model for the case in which the thickness of the plate tends to 0. This model contains membrane and plate deformations which are coupled via plastic strains. The convergence analysis is based on an abstract Gamma convergence theory for rate-independent evolution formulated in the framework of energetic solutions. This concept is based on an energy-storage functional and a dissipation functional, such that the notion of solution is phrased in terms of a stability condition and an energy balance.

S. Arnrich, A. Mielke, M.A. Peletier, G. Savaré, M. Veneroni, Passing to the limit in a Wasserstein gradient flow: From diffusion to reaction, Calculus of Variations and Partial Differential Equations, 44 (2012), pp. 419--454.AbstractWe study a singular-limit problem arising in the modelling of chemical reactions. At finite $e>0$, the system is described by a Fokker-Planck convection-diffusion equation with a double-well convection potential. This potential is scaled by $1/e$, and in the limit $eto0$, the solution concentrates onto the two wells, resulting into a limiting system that is a pair of ordinary differential equations for the density at the two wells. This convergence has been proved in Peletier, Savaré, and Veneroni, em SIAM Journal on Mathematical Analysis, 42(4):1805--1825, 2010, using the linear structure of the equation. In this paper we re-prove the result by using solely the Wasserstein gradient-flow structure of the system. In particular, we make no use of the linearity, nor of the fact that it is a second-order system. The first key step in this approach is a reformulation of the equation as the minimization of an action functional that captures the propety of being a emphcurve of maximal slope in an integrated form. The second important step is a rescaling of space. Using only the Wasserstein gradient-flow structure, we prove that the sequence of rescaled solutions is pre-compact in an appropriate topology. We then prove a Gamma-convergence result for the functional in this topology, and we identify the limiting functional and the differential equation that it represents. A consequence of these results is that solutions of the $e$-problem converge to a solution of the limiting problem.

P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, Continuous dependence for a nonstandard Cahn--Hilliard system with nonlinear atom mobility, Rendiconti del Seminario Matematico. Universita e Politecnico Torino, 70 (2012), pp. 27--52.AbstractThis note is concerned with a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. The system arises from a model of two-species phase segregation on an atomic lattice [Podio-Guidugli 2006]; it consists of the balance equations of microforces and microenergy; the two unknowns are the order parameter $rho$ and the chemical potential $mu$. Some recent results obtained for this class of problems is reviewed and, in the case of a nonconstant and nonlinear atom mobility, uniqueness and continuous dependence on the initial data are shown with the help of a new line of argumentation developed in Colli/Gilardi/Podio-Guidugli/Sprekels 2012.

P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, Distributed optimal control of a nonstandard system of phase field equations, Continuum Mechanics and Thermodynamics, 24 (2012), pp. 437--459.AbstractWe investigate a distributed optimal control problem for a phase field model of Cahn-Hilliard type. The model describes two-species phase segregation on an atomic lattice under the presence of diffusion; it has been introduced recently in [4], on the basis of the theory developed in [15], and consists of a system of two highly nonlinearly coupled PDEs. For this reason, standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional are of standard type. We show that the problem admits a solution, and we derive the first-order necessary conditions of optimality.

P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, Global existence for a strongly coupled Cahn--Hilliard system with viscosity, Bollettino della Unione Matematica Italiana. Serie 9, 5 (2012), pp. 495--513.AbstractAn existence result is proved for a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. This system is meant to model two-species phase segregation on an atomic lattice under the presence of diffusion. A similar system has been recently introduced and analyzed in [CGPS11]. Both systems conform to the general theory developed in [Pod06]: two parabolic PDEs, interpreted as balances of microforces and microenergy, are to be solved for the order parameter $rho$ and the chemical potential $mu$. In the system studied in this note, a phase-field equation in $rho$ fairly more general than in [CGPS11] is coupled with a highly nonlinear diffusion equation for $mu$, in which the conductivity coefficient is allowed to depend nonlinearly on both variables.

P. Colli, G. Gilardi, J. Sprekels, Analysis and optimal boundary control of a nonstandard system of phase field equations, Milan Journal of Mathematics, 80 (2012), pp. 119--149.AbstractWe investigate a nonstandard phase field model of Cahn-Hilliard type. The model, which was introduced in Podio-Guidugli (2006), describes two-species phase segregation and consists of a system of two highly nonlinearly coupled PDEs. It has been studied recently in Colli, Gilardi, Podio-Guidugli, and Sprekels (2011a and b) for the case of homogeneous Neumann boundary conditions. In this paper, we investigate the case that the boundary condition for one of the unknowns of the system is of third kind and nonhomogeneous. For the resulting system, we show well-posedness, and we study optimal boundary control problems. Existence of optimal controls is shown, and the first-order necessary optimality conditions are derived. Owing to the strong nonlinear couplings in the PDE system, standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional will be of standard type.

K. Hackl, S. Heinz, A. Mielke, A model for the evolution of laminates in finite-strain elastoplasticity, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 92 (2012), pp. 888--909.AbstractWe study the time evolution in elastoplasticity within the rate-independent framework of generalized standard materials. Our particular interest is the formation and the evolution of microstructure. Providing models where existence proofs are possible is a challenging task since the presence of microstructure comes along with a lack of convexity and, hence, compactness arguments cannot be applied to prove the existence of solutions. In order to overcome this problem, we will incorporate information on the microstructure into the internal variable, which is still compatible with generalized standard materials. More precisely, we shall allow for such microstructure that is given by simple or sequential laminates. We will consider a model for the evolution of these laminates and we will prove a theorem on the existence of solutions to any finite sequence of time-incremental minimization problems. In order to illustrate the mechanical consequences of the theory developed some numerical results, especially dealing with the rotation of laminates, are presented.

A.F.M. TERElst, J. Rehberg, $L^infty$-estimates for divergence operators on bad domains, Analysis and Applications, 10 (2012), pp. 207--214.AbstractIn this paper, we prove $L^infty$-estimates for solutions of divergence operators in case of mixed boundary conditions. In this very general setting, the Dirichlet boundary part may be arbitrarily wild, i.e. no regularity conditions have to be imposed on it.

G. Aki, J. Daube, W. Dreyer, J. Giesselmann, M. Kränkel, Ch. Kraus, A diffuse interface model for quasi-incompressible flows: Sharp interface limits and numerics, ESAIM Proceedings, 38 (2012), pp. 54--77.AbstractIn this contribution, we investigate a diffuse interface model for quasi-incompressible flows. We determine corresponding sharp interface limits of two different scalings. The sharp interface limit is deduced by matched asymptotic expansions of the fields in powers of the interface. In particular, we study solutions of the derived system of inner equations and discuss the results within the general setting of jump conditions for sharp interface models. Furthermore, we treat, as a subproblem, the convective Cahn-Hilliard equation numerically by a Local Discontinuous Galerkin scheme.

A. Glitzky, An electronic model for solar cells including active interfaces and energy resolved defect densities, SIAM Journal on Mathematical Analysis, 44 (2012), pp. 3874--3900.AbstractWe introduce an electronic model for solar cells taking into account heterostructures with active interfaces and energy resolved volume and interface trap densities. The model consists of continuity equations for electrons and holes with thermionic emission transfer conditions at the interface and of ODEs for the trap densities with energy level and spatial position as parameters, where the right hand sides contain generation-recombination as well as ionization reactions. This system is coupled with a Poisson equation for the electrostatic potential. We show the thermodynamic correctness of the model and prove a priori estimates for the solutions to the evolution system. Moreover, existence and uniqueness of weak solutions of the problem are proven. For this purpose we solve a regularized problem and verify bounds of the corresponding solution not depending on the regularization level.

D. Knees, A. Schröder, Global spatial regularity for elasticity models with cracks, contact and other nonsmooth constraints, Mathematical Methods in the Applied Sciences, 35 (2012), pp. 1859--1884.AbstractA global higher differentiability result in Besov spaces is proved for the displacement fields of linear elastic models with self contact. Domains with cracks are studied, where nonpenetration conditions/Signorini conditions are imposed on the crack faces. It is shown that in a neighborhood of crack tips (in 2D) or crack fronts (3D) the displacement fields are B 3/2 2,∞ regular. The proof relies on a difference quotient argument for the directions tangential to the crack. In order to obtain the regularity estimates also in the normal direction, an argument due to Ebmeyer/Frehse/Kassmann is modified. The methods are then applied to further examples like contact problems with nonsmooth rigid foundations, to a model with Tresca friction and to minimization problems with nonsmooth energies and constraints as they occur for instance in the modeling of shape memory alloys. Based on Falk's approximation Theorem for variational inequalities, convergence rates for FE-discretizations of contact problems are derived relying on the proven regularity properties. Several numerical examples illustrate the theoretical results.

W. Dreyer, J. Giesselmann, Ch. Kraus, Ch. Rohde, Asymptotic analysis for Korteweg models, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 14 (2012), pp. 105--143.AbstractThis paper deals with a sharp interface limit of the isothermal Navier-Stokes-Korteweg system. The sharp interface limit is performed by matched asymptotic expansions of the fields in powers of the interface width. These expansions are considered in the interfacial region (inner expansions) and in the bulk (outer expansion) and are matched order by order. Particularly we consider the first orders of the corresponding inner equations obtained by a change of coordinates in an interfacial layer. For a specific scaling we establish solvability criteria for these inner equations and recover the results within the general setting of jump conditions for sharp interface models.

A. Mielke, R. Rossi, G. Savaré, Variational convergence of gradient flows and rate-independent evolutions in metric spaces, Milan Journal of Mathematics, 80 (2012), pp. 381--410.AbstractWe study the asymptotic behaviour of families of gradient flows in a general metric setting, when the metric-dissipation potentials degenerate in the limit to a dissipation with linear growth. We present a general variational definition of BV solutions to metric evolutions, showing the different characterization of the solution in the absolutely continuous regime, on the singular Cantor part, and along the jump transitions. By using tools of metric analysis, BV functions and blow-up by time rescaling, we show that this variational notion is stable with respect to a wide class of perturbations involving energies, distances, and dissipation potentials. As a particular application, we show that BV solutions to rate-independent problems arise naturally as a limit of p-gradient flows, p>1, when the exponents p converge to 1.

A. Mielke, Emergence of rate-independent dissipation from viscous systems with wiggly energies, Continuum Mechanics and Thermodynamics, 24 (2012), pp. 591--606.AbstractWe consider the passage from viscous system to rate-independent system in the limit of vanishing viscosity and for wiggly energies. Our new convergence approach is based on the (R,R*) formulation by De Giorgi, where we pass to the Γ limit in the dissipation functional. The difficulty is that the type of dissipation changes from a quadratic functional to one that is homogeneous of degree 1. The analysis uses the decomposition of the restoring force into a macroscopic part and a fluctuating part, where the latter is handled via homogenization.

CH. Heinemann, Ch. Kraus, Existence of weak solutions for Cahn--Hilliard systems coupled with elasticity and damage, Advances in Mathematical Sciences and Applications, 21 (2011), pp. 321--359.AbstractThe Cahn-Hilliard model is a typical phase field approach for describing phase separation and coarsening phenomena in alloys. This model has been generalized to the so-called Cahn-Larché system by combining it with elasticity to capture non-neglecting deformation phenomena, which occurs during phase separation in the material. Evolving microstructures such as phase separation and coarsening processes have a strong influence on damage initiation and propagation in alloys. We develop the existing framework of Cahn-Hilliard and Cahn-Larché systems by coupling these systems with a unidirectional evolution inclusion for an internal variable, describing damage processes. After establishing a weak notion of the corresponding evolutionary system, we prove existence of weak solutions for rate-dependent damage processes under certain growth conditions of the energy functional.

P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, Well-posedness and long-time behavior for a nonstandard viscous Cahn--Hilliard system, SIAM Journal on Applied Mathematics, 71 (2011), pp. 1849--1870.AbstractWe study a diffusion model of phase field type, consisting of a system of two partial differential equations encoding the balances of microforces and microenergy; the two unknowns are the order parameter and the chemical potential. By a careful development of uniform estimates and the deduction of certain useful boundedness properties, we prove existence and uniqueness of a global-in-time smooth solution to the associated initial/boundary-value problem; moreover, we give a description of the relative $omega$-limit set.

R. Haller-Dintelmann, H.-Chr. Kaiser, J. Rehberg, Direct computation of elliptic singularities across anisotropic, multi-material edges, Journal of Mathematical Sciences (New York), 172 (2011), pp. 589--622.AbstractWe characterise the singularities of elliptic div-grad operators at points or edges where several materials meet on a Dirichlet or Neumann part of the boundary of a two- or three-dimensional domain. Special emphasis is put on anisotropic coefficient matrices. The singularities can be computed as roots of a characteristic transcendental equation. We establish uniform bounds for the singular values for several classes of three- and four-material edges. These bounds can be used to prove optimal regularity results for elliptic div-grad operators on three-dimensional, heterogeneous, polyhedral domains with mixed boundary conditions. We demonstrate this for the benchmark L--shape problem.

K. Hermsdörfer, Ch. Kraus, D. Kröner, Interface conditions for limits of the Navier--Stokes--Korteweg model, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 13 (2011), pp. 239--254.AbstractIn this contribution we will study the behaviour of the pressure across phase boundaries in liquid-vapour flows. As mathematical model we will consider the static version of the Navier-Stokes-Korteweg model which belongs to the class of diffuse interface models. From this static equation a formula for the pressure jump across the phase interface can be derived. If we perform then the sharp interface limit we see that the resulting interface condition for the pressure seems to be inconsistent with classical results of hydrodynamics. Therefore we will present two approaches to recover the results of hydrodynamics in the sharp interface limit at least for special situations.

A. Glitzky, Analysis of electronic models for solar cells including energy resolved defect densities, Mathematical Methods in the Applied Sciences, 34 (2011), pp. 1980--1998.AbstractWe introduce an electronic model for solar cells including energy resolved defect densities. The resulting drift-diffusion model corresponds to a generalized van Roosbroeck system with additional source terms coupled with ODEs containing space and energy as parameters for all defect densities. The system has to be considered in heterostructures and with mixed boundary conditions from device simulation. We give a weak formulation of the problem. If the boundary data and the sources are compatible with thermodynamic equilibrium the free energy along solutions decays monotonously. In other cases it may be increasing, but we estimate its growth. We establish boundedness and uniqueness results and prove the existence of a weak solution. This is done by considering a regularized problem, showing its solvability and the boundedness of its solutions independent of the regularization level.

CH. Kraus, The degenerate and non-degenerate Stefan problem with inhomogeneous and anisotropic Gibbs--Thomson law, European Journal of Applied Mathematics, 22 (2011), pp. 393--422.AbstractThe Stefan problem is coupled with a spatially inhomogeneous and anisotropic Gibbs-Thomson condition at the phase boundary. We show the long-time existence of weak solutions for the non-degenerate Stefan problem with a spatially inhomogeneous and anisotropic Gibbs-Thomson law and a conditional existence result for the corresponding degenerate Stefan problem. To this end approximate solutions are constructed by means of variational functionals with spatially inhomogeneous and anisotropic interfacial energy. By passing to the limit, we establish solutions of the Stefan problem with a spatially inhomogeneous and anisotropic Gibbs-Thomson law in a weak generalized BV-formulation.

A. Mielke, U. Stefanelli, Weighted energy-dissipation functionals for gradient flows, ESAIM. Control, Optimisation and Calculus of Variations, 17 (2011), pp. 52--85.AbstractWe investigate a global-in-time variational approach to abstract evolution by means of the weighted energy-dissipation functionals proposed by Mielke & Ortiz in “A class of minimum principles for characterizing the trajectories of dissipative systems”. In particular, we focus on gradient flows in Hilbert spaces. The main result is the convergence of minimizers and approximate minimizers of these functionals to the unique solution of the gradient flow. Sharp convergence rates are provided and the convergence analysis is combined with time-discretization. Applications of the theory to various classes of parabolic PDE problems are presented. In particular, we focus on two examples of microstructure evolution from S. Conti and M. Ortiz “Minimum principles for the trajectories of systems governed by rate problems”.

A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Nonlinearity, 24 (2011), pp. 1329--1346.AbstractIn recent years the theory of Wasserstein distances has opened up a new treatment of the diffusion equations as gradient systems, where the entropy takes the role of the driving functional and where the space is equipped with the Wasserstein metric. We show that this structure can be generalized to closed reaction-diffusion systems, where the free energy (or the entropy) is the driving functional and further conserved quantities may exists, like the total number of chemical species. The metric is constructed by using the dual dissipation potential, which is a convex function of the chemical potentials. In particular, it is possible to treat diffusion and reaction terms simultaneously. The same ideas extend to semiconductor equations involving the electron and hole densities, the electrostatic potential, and the temperature.

A. Mielke, Complete-damage evolution based on energies and stresses, Discrete and Continuous Dynamical Systems -- Series S, 4 (2011), pp. 423--439.AbstractThe rate-independent damage model recently developed in Bouchitté, Mielke, Roubíček “A complete-damage problem at small strains" allows for complete damage, such that the deformation is no longer well-defined. The evolution can be described in terms of energy densities and stresses. Using concepts of parametrized Gamma convergence, we generalize the theory to convex, but non-quadratic elastic energies by providing Gamma convergence of energetic solutions from partial to complete damage under rather general conditions.

A. Glitzky, Uniform exponential decay of the free energy for Voronoi finite volume discretized reaction-diffusion systems, Mathematische Nachrichten, 284 (2011), pp. 2159--2174.AbstractOur focus are energy estimates for discretized reaction-diffusion systems for a finite number of species. We introduce a discretization scheme (Voronoi finite volume in space and fully implicit in time) which has the special property that it preserves the main features of the continuous systems, namely positivity, dissipativity and flux conservation. For a class of Voronoi finite volume meshes we investigate thermodynamic equilibria and prove for solutions to the evolution system the monotone and exponential decay of the discrete free energy to its equilibrium value with a unified rate of decay for this class of discretizations. The essential idea is an estimate of the free energy by the dissipation rate which is proved indirectly by taking into account sequences of Voronoi finite volume meshes. Essential ingredient in that proof is a discrete Sobolev-Poincaré inequality.

J.A. Griepentrog, L. Recke, Local existence, uniqueness, and smooth dependence for nonsmooth quasilinear parabolic problems, Journal of Evolution Equations, 10 (2010), pp. 341--375.AbstractA general theory on local existence, uniqueness, regularity, and smooth dependence in Hölder spaces for a general class of quasilinear parabolic initial boundary value problems with nonsmooth data has been developed. As a result the gap between low smoothness of the data, which is typical for many applications, and high smoothness of the solutions, which is necessary for the applicability of differential calculus to the abstract formulations of the initial boundary value problems, has been closed. The main tools are new maximal regularity results of the first author in Sobolev-Morrey spaces, linearization techniques and the Implicit Function Theorem. Typical applications are transport processes of charged particles in semiconductor heterostructures, phase separation processes of nonlocally interacting particles, chemotactic aggregation in heterogeneous environments as well as optimal control by means of quasilinear elliptic and parabolic PDEs with nonsmooth data.

M. Thomas, A. Mielke, Damage of nonlinearly elastic materials at small strain --- Existence and regularity results, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 90 (2010), pp. 88--112.AbstractIn this paper an existence result for energetic solutions of rate-independent damage processes is established and the temporal regularity of the solution is discussed. We consider a body consisting of a physically nonlinearly elastic material undergoing small deformations and partial damage. The present work is a generalization of [Mielke-Roubicek 2006] concerning the properties of the stored elastic energy density as well as the suitable Sobolev space for the damage variable: While previous work assumes that the damage variable z satisfies z ∈ W^1,r (Omega) with r>d for Omega ⊂ R^d, we can handle the case r>1 by a new technique for the construction of joint recovery sequences. Moreover, this work generalizes the temporal regularity results to physically nonlinearly elastic materials by analyzing Lipschitz- and Hölder-continuity of solutions with respect to time.

H. Garcke, Ch. Kraus, An anisotropic, inhomogeneous, elastically modified Gibbs--Thomson law as singular limit of a diffuse interface model, Advances in Mathematical Sciences and Applications, 20 (2010), pp. 511--545.AbstractWe consider the sharp interface limit of a diffuse phase field model with prescribed total mass taking into account a spatially inhomogeneous anisotropic interfacial energy and an elastic energy. The main aim is the derivation of a weak formulation of an anisotropic, inhomogeneous, elastically modified Gibbs-Thomson law in the sharp interface limit. To this end we show that one can pass to the limit in the weak formulation of the Euler-Lagrange equation of the diffuse phase field energy.

J. Giannoulis, A. Mielke, Ch. Sparber, High-frequency averaging in semi-classical Hartree-type equations, Asymptotic Analysis, 70 (2010), pp. 87--100.AbstractWe investigate the asymptotic behavior of solutions to semi-classical Schröodinger equations with nonlinearities of Hartree type. For a weakly nonlinear scaling, we show the validity of an asymptotic superposition principle for slowly modulated highly oscillatory pulses. The result is based on a high-frequency averaging effect due to the nonlocal nature of the Hartree potential, which inhibits the creation of new resonant waves. In the proof we make use of the framework of Wiener algebras.

P. Gruber, D. Knees, S. Nesenenko, M. Thomas, Analytical and numerical aspects of time-dependent models with internal variables, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 90 (2010), pp. 861--902.AbstractIn this paper some analytical and numerical aspects of time-dependent models with internal variables are discussed. The focus lies on elasto/visco-plastic models of monotone type arising in the theory of inelastic behavior of materials. This class of problems includes the classical models of elasto-plasticity with hardening and viscous models of the Norton-Hoff type. We discuss the existence theory for different models of monotone type, give an overview on spatial regularity results for solutions to such models and illustrate a numerical solution algorithm at an example. Finally, the relation to the energetic formulation for rate-independent processes is explained and temporal regularity results based on different convexity assumptions are presented.

R. Haller-Dintelmann, J. Rehberg, Coercivity for elliptic operators and positivity of solutions on Lipschitz domains, Archiv der Mathematik, 95 (2010), pp. 457--468.AbstractWe show that usual second order operators in divergence form satisfy coercivity on Lipschitz domains if they are either complemented with homogeneous Dirichlet boundary conditions on a set of non-zero boundary measure or if a suitable Robin boundary condition is posed. Moreover, we prove the positivity of solutions in a general, abstract setting, provided that the right hand side is a positive functional. Finally, positive elements from $W^-1,2$ are identified as positive measures.

W. Dreyer, Ch. Kraus, On the van der Waals--Cahn--Hilliard phase-field model and its equilibria conditions in the sharp interface limit, Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, 140 A (2010), pp. 1161--1186.AbstractWe study the equilibria of liquid--vapor phase transitions of a single substance at constant temperature and relate the sharp interface model of classical thermodynamics to a phase field model that determines the equilibria by the stationary van der Waals--Cahn--Hilliard theory. For two reasons we reconsider this old problem. 1. Equilibria in a two phase system can be established either under fixed total volume of the system or under fixed external pressure. The latter case implies that the domain of the two--phase system varies. However, in the mathematical literature rigorous sharp interface limits of phase transitions are usually considered under fixed volume. This brings the necessity to extend the existing tools for rigorous sharp interface limits to changing domains since in nature most processes involving phase transitions run at constant pressure. 2. Thermodynamics provides for a single substance two jump conditions at the sharp interface, viz. the continuity of the specific Gibbs free energies of the adjacent phases and the discontinuity of the corresponding pressures, which is balanced by the mean curvature. The existing estimates for rigorous sharp interface limits show only the first condition. We identify the cause of this phenomenon and develop a strategy that yields both conditions up to the first order. The necessary information on the equilibrium conditions are achieved by an asymptotic expansion of the density which is valid for an arbitrary double well potential. We establish this expansion by means of local energy estimates, uniform convergence results of the density and estimates on the Laplacian of the density.

A. Glitzky, K. Gärtner, Existence of bounded steady state solutions to spin-polarized drift-diffusion systems, SIAM Journal on Mathematical Analysis, 41 (2010), pp. 2489--2513.AbstractWe study a stationary spin-polarized drift-diffusion model for semiconductor spintronic devices. This coupled system of continuity equations and a Poisson equation with mixed boundary conditions in all equations has to be considered in heterostructures. In 3D we prove the existence and boundedness of steady states. If the Dirichlet conditions are compatible or nearly compatible with thermodynamic equilibrium the solution is unique. The same properties are obtained for a space discretized version of the problem: Using a Scharfetter-Gummel scheme on 3D boundary conforming Delaunay grids we show existence, boundedness and, for small applied voltages, the uniqueness of the discrete solution.

D. Hömberg, Ch. Meyer, J. Rehberg, W. Ring, Optimal control for the thermistor problem, SIAM Journal on Control and Optimization, 48 (2010), pp. 3449--3481.AbstractThis paper is concerned with the state-constrained optimal control of the two-dimensional thermistor problem, a quasi-linear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Existence, uniqueness and continuity for the state system are derived by employing maximal elliptic and parabolic regularity. By similar arguments the linearized state system is discussed, while the adjoint system involving measures is investigated using a duality argument. These results allow to derive first-order necessary conditions for the optimal control problem.

H.-Chr. Kaiser, J. Rehberg, Optimal elliptic regularity at the crossing of a material interface and a Neumann boundary edge, Journal of Mathematical Sciences (New York), 169 (2010), pp. 145--166.AbstractWe investigate optimal elliptic regularity of anisotropic div-grad operators in three dimensions at the crossing of a material interface and an edge of the spatial domain on the Neumann boundary part within the scale of Sobolev spaces.

A. Mielke, T. Roubíček, J. Zeman, Complete damage in elastic and viscoelastic media and its energetics, Computer Methods in Applied Mechanics and Engineering, 199 (2010), pp. 1242--1253.AbstractA model for the evolution of damage that allows for complete disintegration is addressed. Small strains and a linear response function are assumed. The “flow rule” for the damage parameter is rate-independent. The stored energy involves the gradient of the damage variable, which determines an internal length-scale. Quasi-static fully rate-independent evolution is considered as well as rate-dependent evolution including viscous/inertial effects. Illustrative 2-dimensional computer simulations are presented, too.

K. Hoke, H.-Chr. Kaiser, J. Rehberg, Analyticity for some operator functions from statistical quantum mechanics, Annales Henri Poincare. A Journal of Theoretical and Mathematical Physics, 10 (2009), pp. 749--771.AbstractFor rather general thermodynamic equilibrium distribution functions the density of a statistical ensemble of quantum mechanical particles depends analytically on the potential in the Schrödinger operator describing the quantum system. A key to the proof is that the resolvent to a power less than one of an elliptic operator with non-smooth coefficients, and mixed Dirichlet/Neumann boundary conditions on a bounded up to three-dimensional Lipschitz domain factorizes over the space of essentially bounded functions.

A. Petrov, M. Schatzman, Mathematical results on existence for viscoelastodynamic problems with unilateral constraints, SIAM Journal on Mathematical Analysis, 40 (2009), pp. 1882--1904.AbstractWe study a damped wave equation and the evolution of a Kelvin-Voigt material, both problems have unilateral boundary conditions. Under appropriate regularity assumptions on the initial data, both problems possess a weak solution which is obtained as the limit of a sequence of penalized problems; the functional properties of all the traces are precisely identified through Fourier analysis, and this enables us to infer the existence of a strong solution.

G. Bouchitté, A. Mielke, T. Roubíček, A complete-damage problem at small strains, Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 60 (2009), pp. 205--236.AbstractThe complete damage of a linearly-responding material that can thus completely disintegrate is addressed at small strains under time-varying Dirichlet boundary conditions as a rate-independent evolution problem in multidimensional situations. The stored energy involves the gradient of the damage variable. This variable as well as the stress and energies are shown to be well defined even under complete damage, in contrast to displacement and strain. Existence of an energetic solution is proved, in particular, by detailed investigating the $Gamma$-limit of the stored energy and its dependence on boundary conditions. Eventually, the theory is illustrated on a one-dimensional example.

R. Haller-Dintelmann, Ch. Meyer, J. Rehberg, A. Schiela, Hölder continuity and optimal control for nonsmooth elliptic problems, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 60 (2009), pp. 397--428.AbstractThe well known De Giorgi result on Hölder continuity for solutions of the Dirichlet problem is re-established for mixed boundary value problems, provided that the underlying domain is a Lipschitz domain and the border between the Dirichlet and the Neumann boundary part satisfies a very general geometric condition. Implications of this result for optimal control theory are presented.

A. Mainik, A. Mielke, Global existence for rate-independent gradient plasticity at finite strain, Journal of Nonlinear Science, 19 (2009), pp. 221--248.AbstractWe provide a global existence result for the time-continuous elastoplasticity problem using the energetic formulation. For this we show that the geometric nonlinearities via the multiplicative decomposition of the strain can be controlled via polyconvexity and a priori stress bounds in terms of the energy density. While temporal oscillations are controlled via the energy dissipation the spatial compactness is obtain via the regularizing terms involving gradients of the internal variables.

A. Glitzky, K. Gärtner, Energy estimates for continuous and discretized electro-reaction-diffusion systems, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 70 (2009), pp. 788--805.AbstractWe consider electro-reaction-diffusion systems consisting of continuity equations for a finite number of species coupled with a Poisson equation. We take into account heterostructures, anisotropic materials and rather general statistic relations. We investigate thermodynamic equilibria and prove for solutions to the evolution system the monotone and exponential decay of the free energy to its equilibrium value. Here the essential idea is an estimate of the free energy by the dissipation rate which is proved indirectly. The same properties are shown for an implicit time discretized version of the problem. Moreover, we provide a space discretized scheme for the electro-reaction-diffusion system which is dissipative (the free energy decays monotonously). On a fixed grid we use for each species different Voronoi boxes which are defined with respect to the anisotropy matrix occurring in the flux term of this species.

A. Glitzky, Energy estimates for electro-reaction-diffusion systems with partly fast kinetics, Discrete and Continuous Dynamical Systems, 25 (2009), pp. 159--174.AbstractWe start from a basic model for the transport of charged species in heterostructures containing the mechanisms diffusion, drift and reactions in the domain and at its boundary. Considering limit cases of partly fast kinetics we derive reduced models. This reduction can be interpreted as some kind of projection scheme for the weak formulation of the basic electro--reaction--diffusion system. We verify assertions concerning invariants and steady states and prove the monotone and exponential decay of the free energy along solutions to the reduced problem and to its fully implicit discrete-time version by means of the results of the basic problem. Moreover we make a comparison of prolongated quantities with the solutions to the basic model.

H.-Chr. Kaiser, H. Neidhardt, J. Rehberg, Classical solutions of drift-diffusion equations for semiconductor devices: The 2D case, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 71 (2009), pp. 1584--1605.AbstractWe regard drift-diffusion equations for semiconductor devices in Lebesgue spaces. To that end we reformulate the (generalized) van Roosbroeck system as an evolution equation for the potentials to the driving forces of the currents of electrons and holes. This evolution equation falls into a class of quasi-linear parabolic systems which allow unique, local in time solution in certain Lebesgue spaces. In particular, it turns out that the divergence of the electron and hole current is an integrable function. Hence, Gauss' theorem applies, and gives the foundation for space discretization of the equations by means of finite volume schemes. Moreover, the strong differentiability of the electron and hole density in time is constitutive for the implicit time discretization scheme. Finite volume discretization of space, and implicit time discretization are accepted custom in engineering and scientific computing. ---This investigation puts special emphasis on non-smooth spatial domains, mixed boundary conditions, and heterogeneous material compositions, as required in electronic device simulation.

H.-Chr. Kaiser, H. Neidhardt, J. Rehberg, Monotonicity properties of the quantum mechanical particle density: An elementary proof, Monatshefte fur Mathematik, 158 (2009), pp. 179--185.AbstractAn elementary proof of the anti-monotonicity of the quantum mechanical particle density with respect to the potential in the Hamiltonian is given for a large class of admissible thermodynamic equilibrium distribution functions. In particular the zero temperature case is included.

D. Knees, On global spatial regularity in elasto-plasticity with linear hardening, Calculus of Variations and Partial Differential Equations, 36 (2009), pp. 611--625.AbstractWe study the global spatial regularity of solutions of elasto-plastic models with linear hardening. In order to point out the main idea, we consider a model problem on a cube, where we describe Dirichlet and Neumann boundary conditions on the top and the bottom, respectively, and periodic boundary conditions on the remaining faces. Under natural smoothness assumptions on the data we obtain u in L∞((0,T);H3/2-δ(Ω)) for the displacements and z in L∞((0,T);H1/2-δ(Ω)) for the internal variables. The proof is based on a difference quotient technique and a reflection argument.

P. Krejčí, M. Liero, Rate independent Kurzweil processes, Applications of Mathematics, 54 (2009), pp. 117--145.AbstractThe Kurzweil integral technique is applied to a class of rate independent processes with convex energy and discontinuous inputs. We prove existence, uniqueness, and continuous data dependence of solutions in $BV$ spaces. It is shown that in the context of elastoplasticity, the Kurzweil solutions coincide with natural limits of viscous regularizations when the viscosity coefficient tends to zero. The discontinuities produce an additional positive dissipation term, which is not homogeneous of degree one.

A. Mielke, F. Rindler, Reverse approximation of energetic solutions to rate-independent processes, NoDEA. Nonlinear Differential Equations and Applications, 16 (2009), pp. 17--40.AbstractEnergetic solutions to rate-independent processes are usually constructed via time-incremental minimization problems. In this work we show that all energetic solutions can be approximated by incremental problems if we allow approximate minimizers, where the error in minimization has to be of the order of the time step. Moreover, we study sequences of problems where the energy functionals have a Gamma limit.

A. Mielke, R. Rossi, G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces, Discrete and Continuous Dynamical Systems, 25 (2009), pp. 585--615.AbstractRate-independent systems allow for solutions with jumps that need additional modeling. Here we suggest a formulation that arises as limit of viscous regularization of the solutions in the extended state space. Hence, our parametrized metric solutions of a rate-independent system are absolutely continuous mappings from a parameter interval into the extended state space. Jumps appear as generalized gradient flows during which the time is constant. The closely related notion of BV solutions is developed afterwards. Our approach is based on the abstract theory of generalized gradient flows in metric spaces, and comparison with other notions of solutions is given.

A. Mielke, T. Roubíček, Numerical approaches to rate-independent processes and applications in inelasticity, ESAIM: Mathematical Modelling and Numerical Analysis, 43 (2009), pp. 399--429.AbstractA general abstract approximation scheme for rate-independent processes in the energetic formulation is proposed and its convergence is proved under various rather mild data qualifications. The abstract theory is illustrated on several examples: plasticity with isotropic hardening, damage, debonding, magnetostriction, and two models of martensitic transformation in shape-memory alloys.

A. Mielke, L. Paoli, A. Petrov, On existence and approximation for a 3D model of thermally-induced phase transformations in shape-memory alloys, SIAM Journal on Mathematical Analysis, 41 (2009), pp. 1388--1414.AbstractThis paper deals with a three-dimensional model for thermal stress-induced transformations in shape-memory materials. Microstructure, like twined martensites, is described mesoscopically by a vector of internal variables containing the volume fractions of each phase. We assume that the temperature variations are prescribed. The problem is formulated mathematically within the energetic framework of rate-independent processes. An existence result is proved and temporal regularity is obtained in case of uniform convexity. We study also space-time discretizations and establish convergence of these approximations.

H. Neidhardt, V.A. Zagrebnov, Linear non-autonomous Cauchy problems and evolution semigroups, Advances in Differential Equations, 14 (2009), pp. 289--340.AbstractThe paper is devoted to the problem of existence of propagators for an abstract linear non-autonomous evolution Cauchy problem of hyperbolic type in separable Banach spaces. The problem is solved using the so-called evolution semigroup approach which reduces the existence problem for propagators to a perturbation problem of semigroup generators. The results are specified to abstract linear non-autonomous evolution equations in Hilbert spaces where the assumption is made that the domains of the quadratic forms associated with the generators are independent of time. Finally, these results are applied to time-dependent Schrödinger operators with moving point interactions in 1D.

H. Stephan, Modeling of drift-diffusion systems, ZAMP Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 60 (2009), pp. 33--53.AbstractWe derive drift-diffusion systems describing transport processes starting from free energy and equilibrium solutions by a unique method. We include several statistics, heterostructures and cross diffusion. The resulting systems of nonlinear partial differential equations conserve mass and positivity, and have a Lyapunov function (free energy). Using the inverse Hessian as mobility, non-degenerate diffusivity matrices turn out to be diagonal, or - in the case of cross diffusion - even constant.

P.-É. Druet, Weak solutions to a stationary heat equation with nonlocal radiation boundary condition and right-hand side in $L^p$ with $pge 1$, Mathematical Methods in the Applied Sciences, 32 (2008), pp. 135--166.AbstractAccurate modeling of heat transfer in high-temperatures situations requires to account for the effect of heat radiation. In complex applications such as Czochralski's method for crystal growth, in which the conduction radiation heat transfer problem couples to an induction heating problem and to the melt flow problem, we hardly can expect from the mathematical theory that the heat sources will be in a better space than L-1. In such situations, the known results on the unique solvability of the heat conduction problem with surface radiation do not apply, since a right-hand side in L-p with p < 6/5 no longer belongs to the dual of the Banach space in which coercivity is obtained. In this paper, we focus on a stationary heat equation with non-local boundary conditions and right-hand side in L-p with p>=1 arbitrary. Essentially, we construct an approximation procedure and, thanks to new coercivity results, we are able to produce energy estimates that involve only the L-p-norm of the heat-sources, and to pass to the limit.

F. Auricchio, A. Mielke, U. Stefanelli, A rate-independent model for the isothermal quasi-static evolution of shape-memory materials, Mathematical Methods in the Applied Sciences, 18 (2008), pp. 125--164.AbstractThis note addresses a three-dimensional model for isothermal stress-induced transformation in shape-memory polycrystalline materials. We treat the problem within the framework of the energetic formulation of rate-independent processes and investigate existence and continuous dependence issues at both the constitutive relation and quasi-static evolution level. Moreover, we focus on time and space approximation as well as on regularization and parameter asymptotics.

H. Cornean, K. Hoke, H. Neidhardt, P.N. Racec, J. Rehberg, A Kohn--Sham system at zero temperature, Journal of Physics. A. Mathematical and General, 41 (2008), pp. 385304/1--385304/21.AbstractAn one-dimensional Kohn-Sham system for spin particles is considered which effectively describes semiconductor nanostructures and which is investigated at zero temperature. We prove the existence of solutions and derive a priori estimates. For this purpose we find estimates for eigenvalues of the Schrödinger operator with effective Kohn-Sham potential and obtain $W^1,2$-bounds of the associated particle density operator. Afterwards, compactness and continuity results allow to apply Schauder's fixed point theorem. In case of vanishing exchange-correlation potential uniqueness is shown by monotonicity arguments. Finally, we investigate the behavior of the system if the temperature approaches zero.

R. Rossi, A. Mielke, G. Savaré, A metric approach to a class of doubly nonlinear evolution equations and applications, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V, VII (2008), pp. 97--169.AbstractThis paper deals with the analysis of a class of doubly nonlinear evolution equations in the framework of a general metric space. We propose for such equations a suitable metric formulation (which in fact extends the notion of Curve of Maximal Slope for gradient flows in metric spaces, see [5]), and prove the existence of solutions for the related Cauchy problem by means of an approximation scheme by time discretization. Then, we apply our results to obtain the existence of solutions to abstract doubly nonlinear equations in reflexive Banach spaces. The metric approach is also exploited to analyze a class of evolution equations in $L^1$ spaces.

A. Glitzky, R. Hünlich, Stationary solutions to an energy model for semiconductor devices where the equations are defined on different domains, Mathematische Nachrichten, 281 (2008), pp. 1676--1693.AbstractWe discuss a stationary energy model from semiconductor modelling. We accept the more realistic assumption that the continuity equations for electrons and holes have to be considered only in a subdomain $Omega_0$ of the domain of definition $Omega$ of the energy balance equation and of the Poisson equation. Here $Omega_0$ corresponds to the region of semiconducting material, $OmegasetminusOmega_0$ represents passive layers. Metals serving as contacts are modelled by Dirichlet boundary conditions. We prove a local existence and uniqueness result for the two-dimensional stationary energy model. For this purpose we derive a $W^1,p$-regularity result for solutions of systems of elliptic equations with different regions of definition and use the Implicit Function Theorem.

A. Glitzky, Analysis of a spin-polarized drift-diffusion model, Advances in Mathematical Sciences and Applications, 18 (2008), pp. 401--427.AbstractWe introduce a spin-polarized drift-diffusion model for semiconductor spintronic devices. This coupled system of continuity equations and a Poisson equation with mixed boundary conditions in all equations has to be considered in heterostructures. We give a weak formulation of this problem and prove an existence and uniqueness result for the instationary problem. If the boundary data is compatible with thermodynamic equilibrium the free energy along the solution decays monotonously and exponentially to its equilibrium value. In other cases it may be increasing but we estimate its growth. Moreover we give upper and lower estimates for the solution.

A. Glitzky, Exponential decay of the free energy for discretized electro-reaction-diffusion systems, Nonlinearity, 21 (2008), pp. 1989--2009.AbstractOur focus are electro-reaction-diffusion systems consisting of continuity equations for a finite number of species coupled with a Poisson equation. We take into account heterostructures, anisotropic materials and rather general statistical relations. We introduce a discretization scheme (in space and fully implicit in time) using a fixed grid but for each species different Voronoi boxes which are defined with respect to the anisotropy matrix occurring in the flux term of this species. This scheme has the special property that it preserves the main features of the continuous systems, namely positivity, dissipativity and flux conservation. For the discretized electro-reaction-diffusion system we investigate thermodynamic equilibria and prove for solutions to the evolution system the monotone and exponential decay of the free energy to its equilibrium value. The essential idea is an estimate of the free energy by the dissipation rate which is proved indirectly.

D. Knees, P. Neff, Regularity up to the boundary for nonlinear elliptic systems arising in time-incremental infinitesimal elasto-plasticity, SIAM Journal on Mathematical Analysis, 40 (2008), pp. 21--43.AbstractIn this note we investigate the question of higher regularity up to the boundary for quasilinear elliptic systems which origin from the time-discretization of models from infinitesimal elasto-plasticity. Our main focus lies on an elasto-plastic Cosserat model. More specifically we show that the time discretization renders $H^2$-regularity of the displacement and $H^1$-regularity for the symmetric plastic strain $varepsilon_p$ up to the boundary provided the plastic strain of the previous time step is in $H^1$, as well. This result contrasts with classical Hencky and Prandtl-Reuss formulations where it is known not to hold due to the occurrence of slip lines and shear bands. Similar regularity statements are obtained for other regularizations of ideal plasticity like viscosity or isotropic hardening. In the first part we recall the time continuous Cosserat elasto-plasticity problem, provide the update functional for one time step and show various preliminary results for the update functional (Legendre-Hadamard/monotonicity). Using non standard difference quotient techniques we are able to show the higher global regularity. Higher regularity is crucial for qualitative statements of finite element convergence. As a result we may obtain estimates linear in the mesh-width $h$ in error estimates.

D. Knees, A. Mielke, Energy release rate for cracks in finite-strain elasticity, Mathematical Methods in the Applied Sciences, 31 (2008), pp. 501--528.AbstractGriffith's fracture criterion describes in a quasistatic setting whether or not a pre-existing crack in an elastic body is stationary for given external forces. In terms of the energy release rate (ERR), which is the derivative of the deformation energy of the body with respect to a virtual crack extension, this criterion reads: If the ERR is less than a specific constant, then the crack is stationary, otherwise it will grow. In this paper, we consider geometrically nonlinear elastic models with polyconvex energy densities and prove that the ERR is well defined. Moreover, without making any assumption on the smoothness of minimizers, we derive rigorously the well-known Griffith formula and the $J$-integral, from which the ERR can be calculated. The proofs are based on a weak convergence result for Eshelby tensors.

D. Knees, A. Mielke, Ch. Zanini, On the inviscid limit of a model for crack propagation, Mathematical Models & Methods in Applied Sciences, 18 (2008), pp. 1529--1569.AbstractWe study the evolution of a single crack in an elastic body and assume that the crack path is known in advance. The motion of the crack tip is modeled as a rate-independent process on the basis of Griffith's local energy release rate criterion. According to this criterion, the system may stay in a local minimum before it performs a jump. The goal of this paper is to prove existence of such an evolution and to shed light on the discrepancy between the local energy release rate criterion and models which are based on a global stability criterion (as for example the Francfort/Marigo model). We construct solutions to the local model via the vanishing viscosity method and compare different notions of weak, local and global solutions.

A. Mielke, Weak-convergence methods for Hamiltonian multiscale problems, Discrete and Continuous Dynamical Systems, 20 (2008), pp. 53--79.AbstractWe consider Hamiltonian problems depending on a small parameter like in wave equations with rapidly oscillating coefficients or the embedding of an infinite atomic chain into a continuum by letting the atomic distance tend to $0$. For general semilinear Hamiltonian systems we provide abstract convergence results in terms of the existence of a family of joint recovery operators which guarantee that the effective equation is obtained by taking the $Gamma$-limit of the Hamiltonian. The convergence is in the weak sense with respect to the energy norm. Exploiting the well-developed theory of $Gamma$-convergence, we are able to generalize the admissible coefficients for homogenization in the wave equations. Moreover, we treat the passage from a discrete oscillator chain to a wave equation with general $rmL^infty$ coefficients

A. Mielke, A. Petrov, J.A.C. Martins, Convergence of solutions of kinetic variational inequalities in the rate-independent quasi-static limit, Journal of Mathematical Analysis and Applications, 348 (2008), pp. 1012--1020.AbstractThis paper discusses the convergence of kinetic variational inequalities to rate-independent quasi-static variational inequalities. Mathematical formulations as well as existence and uniqueness results for kinetic and rate-independent quasi-static problems are provided. Sharp a priori estimates for the kinetic problem are derived that imply that the kinetic solutions converge to the rate-independent ones, when the size of initial perturbations and the rate of application of the forces tends to 0. An application to three-dimensional elastic-plastic systems with hardening is given.

J.A. Griepentrog, Maximal regularity for nonsmooth parabolic problems in Sobolev--Morrey spaces, Advances in Differential Equations, 12 (2007), pp. 1031--1078.AbstractThis text is devoted to maximal regularity results for second order parabolic systems on LIPSCHITZ domains of space dimension greater or equal than three with diagonal principal part, nonsmooth coefficients, and nonhomogeneous mixed boundary conditions. We show that the corresponding class of initial boundary value problems generates isomorphisms between two scales of SOBOLEV-MORREY spaces for solutions and right hand sides introduced in the first part of our presentation. The solutions depend smoothly on the data of the problem. Moreover, they are HOELDER continuous in time and space up to the boundary for a certain range of MORREY exponents. Due to the complete continuity of embedding and trace maps these results remain true for a broad class of unbounded lower order coefficients.

J.A. Griepentrog, Sobolev--Morrey spaces associated with evolution equations, Advances in Differential Equations, 12 (2007), pp. 781--840.AbstractIn this text we introduce new classes of SOBOLEV-MORREY spaces being adequate for the regularity theory of second order parabolic boundary value problems on LIPSCHITZ domains of space dimension greater or equal than three with nonsmooth coefficients and mixed boundary conditions. We prove embedding and trace theorems as well as invariance properties of these spaces with respect to localization, LIPSCHITZ transformation, and reflection. In the second part of our presentation we show that the class of second order parabolic systems with diagonal principal part generates isomorphisms between the above mentioned SOBOLEV-MORREY spaces of solutions and right hand sides.

A. Mielke, A. Petrov, Thermally driven phase transformation in shape-memory alloys, Advances in Mathematical Sciences and Applications, 17 (2007), pp. 667--685.AbstractThis paper analyzes a model for phase transformation in shape-memory alloys induced by temperature changes and by mechanical loading. We assume that the temperature is prescribed and formulate the problem within the framework of the energetic theory of rate-independent processes. Existence and uniqueness results are proved.

A. Mielke, A. Timofte, Two-scale homogenization for evolutionary variational inequalities via the energetic formulation, SIAM Journal on Mathematical Analysis, 39 (2007), pp. 642--668.AbstractThis paper is devoted to the two-scale homogenization for a class of rate-independent systems described by the energetic formulation or equivalently by an evolutionary variational inequality. In particular, we treat the classical model of linearized elastoplasticity with hardening. The associated nonlinear partial differential inclusion has periodically oscillating coefficients, and the aim is to find a limit problem for the case that the period tends to 0. Our approach is based on the notion of energetic solutions which is phrased in terms of a stability condition and an energy balance of an energy-storage functional and a dissipation functional. Using the recently developed method of weak and strong two-scale convergence via periodic unfolding, we show that these two functionals have a suitable two-scale limit, but now involving the macroscopic variable in the physical domain as well as the microscopic variable in the periodicity cell. Moreover, relying on an abstract theory of Gamma convergence for the energetic formulation using so-called joint recovery sequences it is possible to show that the solutions of the problem with periodicity converge to the energetic solution associated with the limit functionals.

F. Schmid, A. Mielke, Existence results for a contact problem with varying friction coefficient and nonlinear forces, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 87 (2007), pp. 616--631.AbstractWe consider the rate-independent problem of a particle moving in a three - dimensional half space subject to a time-dependent nonlinear restoring force having a convex potential and to Coulomb friction along the flat boundary of the half space, where the friction coefficient may vary along the boundary. Our existence result allows for solutions that may switch arbitrarily often between unconstrained motion in the interior and contact where the solutions may switch between sticking and frictional sliding. However, our existence result is local and guarantees continuous solutions only as long as the convexity of the potential is strong enough to compensate the variation of the friction coefficient times the contact pressure. By simple examples we show that our sufficient conditions are also necessary. Our method is based on the energetic formulation of rate-independent systems as developed by Mielke and co-workers. We generalize the time-incremental minimization procedure of Mielke and Rossi for the present situation of a non-associative flow rule.

M. Baro, H.-Chr. Kaiser, H. Neidhardt, J. Rehberg, A quantum transmitting Schrödinger-Poisson system, Reviews in Mathematical Physics. A Journal for Both Review and Original Research Papers in the Field of Mathematical Physics, 16 (2004), pp. 281--330.

J.A. Griepentrog, An application of the Implicit Function Theorem to an energy model of the semiconductor theory, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 79 (1999), pp. 43--51.AbstractIn this paper we deal with a mathematical model for the description of heat conduction and carrier transport in semiconductor heterostructures. We solve a coupled system of nonlinear elliptic differential equations consisting of the heat equation with Joule heating as a source, the Poisson equation for the electric field an drift-diffusion equations with temperature dependent coefficients describing the charge and current conservation, subject to general thermal and electrical boundary conditions. We prove the existence and uniqueness of Holder continuous weak solutions near thermodynamic equilibria points using the Implicit Function Theorem. To show the differentiability of maps corresponding to the weak formulation of the problem we use regularity results from the theory of nonsmooth linear elliptic boundary value problems in Sobolev-Campanato spaces.

Beiträge zu Sammelwerken

P. Colli, G. Gilardi, J. Sprekels, Nonlocal phase field models of viscous Cahn--Hilliard type, in: Topics in Applied Analysis and Optimisation, M. Hintermüller, J.F. Rodrigues, eds., CIM Series in Mathematical Sciences, Springer Nature Switzerland AG, Cham, 2019, pp. 71--100, DOI 10.1007/978-3-030-33116-0 .AbstractA nonlocal phase field model of viscous Cahn--Hilliard type is considered. This model constitutes a nonlocal version of a model for two-species phase segregation on an atomic lattice under the presence of diffusion that has been studied in a series of papers by P. Podio-Guidugli and the present authors. The resulting system of differential equations consists of a highly nonlinear parabolic equation coupled to a nonlocal ordinary differential equation, which has singular terms that render the analysis difficult. Some results are presented on the well-posedness and stability of the system as well as on the distributed optimal control problem.

S. Bartels, M. Milicevic, M. Thomas, Numerical approach to a model for quasistatic damage with spatial $BV$-regularization, in: Proceedings of the INdAM-ISIMM Workshop on Trends on Applications of Mathematics to Mechanics, Rome, Italy, September 2016, E. Rocca, U. Stefanelli, L. Truskinovsky, A. Visintin, eds., 27 of Springer INdAM Series, Springer International Publishing, Cham, 2018, pp. 179--203, DOI 10.1007/978-3-319-75940-1_9 .AbstractWe address a model for rate-independent, partial, isotropic damage in quasistatic small strain linear elasticity, featuring a damage variable with spatial BV-regularization. Discrete solutions are obtained using an alternate time-discrete scheme and the Variable-ADMM algorithm to solve the constrained nonsmooth optimization problem that determines the damage variable at each time step. We prove convergence of the method and show that discrete solutions approximate a semistable energetic solution of the rate-independent system. Moreover, we present our numerical results for two benchmark problems.

P. Colli, G. Gilardi, J. Sprekels, Limiting problems for a nonstandard viscous Cahn--Hilliard system with dynamic boundary conditions, in: Proceedings of the INdAM-ISIMM Workshop on Trends on Applications of Mathematics to Mechanics, Rome, Italy, September 2016, E. Rocca, U. Stefanelli, L. Truskinovsky, A. Visintin, eds., 27 of Springer INdAM Series, Springer International Publishing, Cham, 2018, pp. 217--242, DOI 10.1007/978-3-319-75940-1_11 .AbstractThis note is concerned with a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by boundary and initial conditions. The system arises from a model of two-species phase segregation on an atomic lattice and was introduced by Podio-Guidugli in Ric. Mat. 55 (2006), pp.105--118. The two unknowns are the phase parameter and the chemical potential. In contrast to previous investigations about this PDE system, we consider here a dynamic boundary condition for the phase variable that involves the Laplace-Beltrami operator and models an additional nonconserving phase transition occurring on the surface of the domain. We are interested to some asymptotic analysis and first discuss the asymptotic limit of the system as the viscosity coefficient of the order parameter equation tends to 0: the convergence of solutions to the corresponding solutions for the limit problem is proven. Then, we study the long-time behavior of the system for both problems, with positive or zero viscosity coefficient, and characterize the omega-limit set in both cases.

M. Thomas, A comparison of delamination models: Modeling, properties, and applications, in: Mathematical Analysis of Continuum Mechanics and Industrial Applications II, Proceedings of the International Conference CoMFoS16, P. VANMeurs, M. Kimura, H. Notsu, eds., 30 of Mathematics for Industry, Springer Nature, Singapore, 2018, pp. 27--38, DOI 10.1007/978-981-10-6283-4_3 .AbstractThis contribution presents recent results in the modeling and the analysis of delamination problems. It addresses adhesive contact, brittle, and cohesive zone models both in a quasistatic and a viscous, dynamic setting for the bulk part. Also different evolution laws for the delaminating surface are discussed.

A. Mielke, Uniform exponential decay for reaction-diffusion systems with complex-balanced mass-action kinetics, in: Patterns of Dynamics, P. Gurevich, J. Hell, B. Sandstede, A. Scheel, eds., Proceedings in Mathematics & Statistics, Springer, 2017, pp. 149--171, DOI 10.1007/978-3-319-64173-7_10 .AbstractWe consider reaction-diffusion systems on a bounded domain with no-flux boundary conditions. All reactions are given by the mass-action law and are assumed to satisfy the complex-balance condition. In the case of a diagonal diffusion matrix, the relative entropy is a Liapunov functional. We give an elementary proof for the Liapunov property as well a few explicit examples for the condition of complex or detailed balancing.

We discuss three methods to obtain energy-dissipation estimates, which guarantee exponential decay of the relative entropy, all of which rely on the log-Sobolev estimate and suitable handling of the reaction terms as well as the mass-conservation relations. The three methods are (i) a convexification argument based on the author's joint work with Haskovec and Markowich, (ii) a series of analytical estimates derived by Desvillettes, Fellner, and Tang, and (iii) a compactness argument of developed by Glitzky and Hünlich.

M. Bulíček, A. Glitzky, M. Liero, Thermistor systems of p(x)-Laplace-type with discontinuous exponents via entropy solutions, in: PDE 2015: Theory and Applications of Partial Differential Equations, H.-Chr. Kaiser, D. Knees, A. Mielke, J. Rehberg, E. Rocca, M. Thomas, E. Valdinoci, eds., 10 of Discrete and Continuous Dynamical Systems, Series S, no. 4, American Institute of Mathematical Sciences, Springfield, 2017, pp. 697--713.AbstractWe show the existence of solutions to a system of elliptic PDEs, that was recently introduced to describe the electrothermal behavior of organic semiconductor devices. Here, two difficulties appear: (i) the elliptic term in the current-flow equation is of p(x)-Laplacian-type with discontinuous exponent p, which limits the use of standard methods, and (ii) in the heat equation, we have to deal with an a priori L1 term on the right hand side describing the Joule heating in the device. We prove the existence of a weak solution under very weak assumptions on the data. Our existence proof is based on Schauder's fixed point theorem and the concept of entropy solutions for the heat equation. Here, the crucial point is the continuous dependence of the entropy solutions on the data of the problem.

P. Colli, J. Sprekels, Optimal boundary control of a nonstandard Cahn--Hilliard system with dynamic boundary condition and double obstacle inclusions, in: Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs: In Honour of Prof. Gianni Gilardi, P. Colli, A. Favini, E. Rocca, G. Schimperna, J. Sprekels, eds., 22 of Springer INdAM Series, Springer International Publishing AG, Cham, 2017, pp. 151--182, DOI 10.1007/978-3-319-64489-9 .AbstractIn this paper, we study an optimal boundary control problem for a model for phase separation taking place in a spatial domain that was introduced by P.Podio-Guidugli in Ric. Mat. 55 (2006), pp.105-118. The model consists of a strongly coupled system of nonlinear parabolic differential inclusions, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically; the system is complemented by initial and boundary conditions. For the order parameter of the phase separation process, a dynamic boundary condition involving the Laplace-Beltrami operator is assumed, which models an additional nonconserving phase transition occurring on the surface of the domain. We complement in this paper results that were established in the recent contribution appeared in Evol. Equ. Control Theory 6 (2017), pp. 35-58, by the two authors and Gianni Gilardi. In contrast to that paper, in which differentiable potentials of logarithmic type were considered, we investigate here the (more difficult) case of nondifferentiable potentials of double obstacle type. For such nonlinearities, the standard techniques of optimal control theory to establish the existence of Lagrange multipliers for the state constraints are known to fail. To overcome these difficulties, we employ the following line of approach: we use the results contained in the preprint arXiv:1609.07046 [math.AP] (2016), pp. 1-30, for the case of (differentiable) logarithmic potentials and perform a so-called "deep quench limit". Using compactness and monotonicity arguments, it is shown that this strategy leads to the desired first-order necessary optimality conditions for the case of (nondifferentiable) double obstacle potentials.

G. Lazzaroni, R. Rossi, M. Thomas, R. Toader, Some remarks on a model for rate-independent damage in thermo-visco-elastodynamics, in: MURPHYS-HSFS-2014: 7th International Workshop on MUlti-Rate Processes and HYSteresis (MURPHYS) & 2nd International Workshop on Hysteresis and Slow-Fast Systems (HSFS), O. Klein, M. Dimian, P. Gurevich, D. Knees, D. Rachinskii, S. Tikhomirov, eds., 727 of Journal of Physics: Conference Series, IOP Publishing, 2016, pp. 012009/1--012009/20.AbstractThis note deals with the analysis of a model for partial damage, where the rate-independent, unidirectional flow rule for the damage variable is coupled with the rate-dependent heat equation, and with the momentum balance featuring inertia and viscosity according to Kelvin-Voigt rheology. The results presented here combine the approach from [Roubíček M2AS'09, SIAM'10] with the methods from Lazzaroni/Rossi/Thomas/Toader [WIAS Preprint 2025]. The present analysis encompasses, differently from [Roubíček SIAM'10], the monotonicity in time of damage and the dependence of the viscous tensor on damage and temperature, and, unlike [WIAS Preprint 2025], a nonconstant heat capacity and a time-dependent Dirichlet loading.

M. Thomas, E. Bonetti, E. Rocca, R. Rossi, A rate-independent gradient system in damage coupled with plasticity via structured strains, in: Gradient Flows: From Theory to Application, B. Düring, C.-B. Schönlieb, M.-Th. Wolfram, eds., 54 of ESAIM Proceedings and Surveys, EDP Sciences, 2016, pp. 54--69.AbstractThis contribution deals with a class of models combining isotropic damage with plasticity. It has been inspired by a work by Freddi and Royer-Carfagni, including the case where the inelastic part of the strain only evolves in regions where the material is damaged. The evolution both of the damage and of the plastic variable is assumed to be rate-independent. Existence of solutions is established in the abstract energetic framework elaborated by Mielke and coworkers.

A. Mielke, R. Rossi, G. Savaré, Balanced-Viscosity solutions for multi-rate systems, in: MURPHYS-HSFS-2014: 7th MUlti-Rate Processes and HYSteresis (MURPHYS) & 2nd International Workshop on Hysteresis and Slow-Fast Systems (HSFS), O. Klein, M. Dimian, P. Gurevich, D. Knees, D. Rachinskii, S. Tikhomirov, eds., 727 of Journal of Physics: Conference Series, IOP Publishing, 2016, pp. 012010/1--012010/26.AbstractSeveral mechanical systems are modeled by the static momentum balance for the displacement u coupled with a rate-independent flow rule for some internal variable z. We consider a class of abstract systems of ODEs which have the same structure, albeit in a finite-dimensional setting, and regularize both the static equation and the rate-independent flow rule by adding viscous dissipation terms with coefficients εα and ε, where 00 is a fixed parameter. Therefore for α different from 1 the variables u and z have different relaxation rates. We address the vanishing-viscosity analysis as ε tends to 0 in the viscous system. We prove that, up to a subsequence, (reparameterized) viscous solutions converge to a parameterized curve yielding a Balanced Viscosity solution to the original rate-independent system and providing an accurate description of the system behavior at jumps. We also give a reformulation of the notion of Balanced Viscosity solution in terms of a system of subdifferential inclusions, showing that the viscosity in u and the one in z are involved in the jump dynamics in different ways, according to whether α >1, α=1, or 0

A. Mielke, On thermodynamical couplings of quantum mechanics and macroscopic systems, in: Mathematical Results in Quantum Mechanics. Proceedings of the QMath12 Conference, P. Exner, W. König, H. Neidhardt, eds., World Scientific Publishing, Singapore, 2015, pp. 331--348.AbstractPure quantum mechanics can be formulated as a Hamiltonian system in terms of the Liouville equation for the density matrix. Dissipative effects are modeled via coupling to a macroscopic system, where the coupling operators act via commutators. Following Öttinger (2010) we use the GENERIC framework to construct thermodynamically consistent evolution equations as a sum of a Hamiltonian and a gradient-flow contribution, which satisfy a particular non-interaction condition: We give three applications of the theory. First, we consider a finite-dimensional quantum system that is coupled to a finite number of simple heat baths, each of which is described by a scalar temperature variable. Second, we model quantum system given by a one-dimensional Schrödinger operator connected to a one-dimensional heat equation on the left and on the right. Finally, we consider thermo-opto-electronics, where the Maxwell-Bloch system of optics is coupled to the energy-drift-diffusion system for semiconductor electronics.

Preprints, Reports, Technical Reports

TH. Eiter, On the spatially asymptotic structure of time-periodic solutions to the Navier--Stokes equations, Preprint no. 2727, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2727 .Abstract, PDF (258 kByte)The asymptotic behavior of weak time-periodic solutions to the Navier--Stokes equations with a drift term in the three-dimensional whole space is investigated. The velocity field is decomposed into a time-independent and a remaining part, and separate asymptotic expansions are derived for both parts and their gradients. One observes that the behavior at spatial infinity is determined by the corresponding Oseen fundamental solutions.

P. Colli, G. Gilardi, J. Sprekels, Optimal control of a phase field system of Caginalp type with fractional operators, Preprint no. 2725, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2725 .Abstract, PDF (360 kByte)In their recent work “Well-posedness, regularity and asymptotic analyses for a fractional phase field system” (Asymptot. Anal. 114 (2019), 93--128), two of the present authors have studied phase field systems of Caginalp type, which model nonconserved, nonisothermal phase transitions and in which the occurring diffusional operators are given by fractional versions in the spectral sense of unbounded, monotone, selfadjoint, linear operators having compact resolvents. In this paper, we complement this analysis by investigating distributed optimal control problems for such systems. It is shown that the associated control-to-state operator is Fréchet differentiable between suitable Banach spaces, and meaningful first-order necessary optimality conditions are derived in terms of a variational inequality and the associated adjoint state variables.

R. Chill, H. Meinlschmidt, J. Rehberg, On the numerical range of second order elliptic operators with mixed boundary conditions in Lp, Preprint no. 2723, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2723 .Abstract, PDF (247 kByte)We consider second order elliptic operators with real, nonsymmetric coefficient functions which are subject to mixed boundary conditions. The aim of this paper is to provide uniform resolvent estimates for the realizations of these operators on Lp in a most direct way and under minimal regularity assumptions on the domain. This is analogous to the main result in [7]. Ultracontractivity of the associated semigroups is also considered. All results are for two different form domains realizing mixed boundary conditions. We further consider the case of Robin- instead of classical Neumann boundary conditions and also allow for operators inducing dynamic boundary conditions. The results are complemented by an intrinsic characterization of elements of the form domains inducing mixed boundary conditions.

J. Sprekels, F. Tröltzsch, Sparse optimal control of a phase field system with singular potentials arising in the modeling of tumor growth, Preprint no. 2721, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2721 .Abstract, PDF (326 kByte)In this paper, we study an optimal control problem for a nonlinear system of reaction-diffusion equations that constitutes a simplified and relaxed version of a thermodynamically consistent phase field model for tumor growth originally introduced in [13]. The model takes the effect of chemotaxis into account but neglects velocity contributions. The unknown quantities of the governing state equations are the chemical potential, the (normalized) tumor fraction, and the nutrient extra-cellular water concentration. The equation governing the evolution of the tumor fraction is dominated by the variational derivative of a double-well potential which may be of singular (e.g., logarithmic) type. In contrast to the recent paper [10] on the same system, we consider in this paper sparsity effects, which means that the cost functional contains a nondifferentiable (but convex) contribution like the L1-norm. For such problems, we derive first-order necessary optimality conditions and conditions for directional sparsity, both with respect to space and time, where the latter case is of particular interest for practical medical applications in which the control variables are given by the administration of cytotoxic drugs or by the supply of nutrients. In addition to these results, we prove that the corresponding control-to-state operator is twice continuously differentiable between suitable Banach spaces, using the implicit function theorem. This result, which complements and sharpens a differentiability result derived in [10], constitutes a prerequisite for a future derivation of second-order sufficient optimality conditions.

D. Bothe, P.-É. Druet, Well-posedness analysis of multicomponent incompressible flow models, Preprint no. 2720, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2720 .Abstract, PDF (465 kByte)In this paper, we extend our study of mass transport in multicomponent isothermal fluids to the incompressible case. For a mixture, incompressibility is defined as the independence of average volume on pressure, and a weighted sum of the partial mass densities of the species stays constant. In this type of models, non solenoidal effects affect the velocity field in the Navier--Stokes equations and, due to different specific volumes of the species, the pressure remains connected to the densities by algebraic formula. By means of a change of variables in the transport problem, we equivalently reformulate the PDE system as to eliminate positivity and incompressibility constraints affecting the density, and prove two type of results: the local-in-time well-posedness in classes of strong solutions, and the global-in-time existence of solutions for initial data sufficiently close to a smooth equilibrium solution.

G. Annegret, M. Liero, G. Nika, Dimension reduction of thermistor models for large-area organic light-emitting diodes, Preprint no. 2719, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2719 .Abstract, PDF (328 kByte)An effective system of partial differential equations describing the heat and current flow through a thin organic light-emitting diode (OLED) mounted on a glass substrate is rigorously derived from a recently introduced fully three-dimensional φ(x)-Laplace thermistor model. The OLED consists of several thin layers that scale differently with respect to the multiscale parameter ε > 0 which is the ratio between the total thickness and the lateral extent of the OLED. Starting point of the derivation is a rescaled formulation of the current-flow equation in the OLED for the driving potential and the heat equation in OLED and glass substrate with Joule heat term concentrated in the OLED. Assuming physically motivated scalings in the electrical flux functions, uniform a priori bounds are derived for the solutions of the three-dimensional system which facilitates the extraction of converging subsequences with limits that are identified as solutions of a dimension reduced system. In the latter, the effective current-flow equation is given by two semilinear equations in the two-dimensional cross-sections of the electrodes and algebraic equations for the continuity of the electrical fluxes through the organic layers. The effective heat equation is formulated only in the glass substrate with Joule heat term on the part of the boundary where the OLED is mounted.

A. Mielke, R.R. Netz, S. Zendehroud, A rigorous derivation and energetics of a wave equation with fractional damping, Preprint no. 2718, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2718 .Abstract, PDF (312 kByte)We consider a linear system that consists of a linear wave equation on a horizontal hypersurface and a parabolic equation in the half space below. The model describes longitudinal elastic waves in organic monolayers at the water-air interface, which is an experimental setup that is relevant for understanding wave propagation in biological membranes. We study the scaling regime where the relevant horizontal length scale is much larger than the vertical length scale and provide a rigorous limit leading to a fractionally-damped wave equation for the membrane. We provide the associated existence results via linear semigroup theory and show convergence of the solutions in the scaling limit. Moreover, based on the energy-dissipation structure for the full model, we derive a natural energy and a natural dissipation function for the fractionally-damped wave equation with a time derivative of order 3/2.

E. Ipocoana, A. Zafferi, Further regularity and uniqueness results for a non-isothermal Cahn--Hilliard equation, Preprint no. 2716, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2716 .Abstract, PDF (270 kByte)The aim of this paper is to establish new regularity results for a non-isothermal Cahn--Hilliard system in the two-dimensional setting. The main achievement is a crucial L∞ estimate for the temperature, obtained by a suitable Moser iteration scheme. Our results in particular allow us to get a new simplified version of the uniqueness proof for the considered model.

J. Maas, A. Mielke, Modeling of chemical reaction systems with detailed balance using gradient structures, Preprint no. 2712, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2712 .Abstract, PDF (552 kByte)We consider various modeling levels for spatially homogeneous chemical reaction systems, namely the chemical master equation, the chemical Langevin dynamics, and the reaction-rate equation. Throughout we restrict our study to the case where the microscopic system satisfies the detailed-balance condition. The latter allows us to enrich the systems with a gradient structure, i.e. the evolution is given by a gradient-flow equation. We present the arising links between the associated gradient structures that are driven by the relative entropy of the detailed-balance steady state. The limit of large volumes is studied in the sense of evolutionary Γ-convergence of gradient flows. Moreover, we use the gradient structures to derive hybrid models for coupling different modeling levels.

S. Bartels, M. Milicevic, M. Thomas, N. Weber, Fully discrete approximation of rate-independent damage models with gradient regularization, Preprint no. 2707, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2707 .Abstract, PDF (3444 kByte)This work provides a convergence analysis of a time-discrete scheme coupled with a finite-element approximation in space for a model for partial, rate-independent damage featuring a gradient regularization as well as a non-smooth constraint to account for the unidirectionality of the damage evolution. The numerical algorithm to solve the coupled problem of quasistatic small strain linear elasticity with rate-independent gradient damage is based on a Variable ADMM-method to approximate the nonsmooth contribution. Space-discretization is based on P1 finite elements and the algorithm directly couples the time-step size with the spatial grid size h. For a wide class of gradient regularizations, which allows both for Sobolev functions of integrability exponent r ∈ (1, ∞) and for BV-functions, it is shown that solutions obtained with the algorithm approximate as h → 0 a semistable energetic solution of the original problem. The latter is characterized by a minimality property for the displacements, a semistability inequality for the damage variable and an energy dissipation estimate. Numerical benchmark experiments confirm the stability of the method.

P.-É. Druet, A theory of generalised solutions for ideal gas mixtures with Maxwell--Stefan diffusion, Preprint no. 2700, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2700 .Abstract, PDF (363 kByte)After the pioneering work by Giovangigli on mathematics of multicomponent flows, several attempts were made to introduce global weak solutions for the PDEs describing the dynamics of fluid mixtures. While the incompressible case with constant density was enlighted well enough due to results by Chen and Jüngel (isothermal case), or Marion and Temam, some open questions remain for the weak solution theory of gas mixtures with their corresponding equations of mixed parabolic-hyperbolic type. For instance, Mucha, Pokorny and Zatorska showed the possibility to stabilise the hyperbolic component by means of the Bresch-Desjardins technique and a regularisation of pressure preventing vacuum. The result by Dreyer, Druet, Gajewski and Guhlke avoids emphex machina stabilisations, but the mathematical assumption that the Onsager matrix is uniformly positive on certain subspaces leads, in the dilute limit, to infinite diffusion velocities which are not compatible with the Maxwell-Stefan form of diffusion fluxes. In this paper, we prove the existence of global weak solutions for isothermal and ideal compressible mixtures with natural diffusion. The main new tool is an asymptotic condition imposed at low pressure on the binary Maxwell-Stefan diffusivities, which compensates possibly extreme behaviour of weak solutions in the rarefied regime.

M. Heida, M. Kantner, A. Stephan, Consistency and convergence for a family of finite volume discretizations of the Fokker--Planck operator, Preprint no. 2684, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2684 .Abstract, PDF (2376 kByte)We introduce a family of various finite volume discretization schemes for the Fokker--Planck operator, which are characterized by different weight functions on the edges. This family particularly includes the well-established Scharfetter--Gummel discretization as well as the recently developed square-root approximation (SQRA) scheme. We motivate this family of discretizations both from the numerical and the modeling point of view and provide a uniform consistency and error analysis. Our main results state that the convergence order primarily depends on the quality of the mesh and in second place on the quality of the weights. We show by numerical experiments that for small gradients the choice of the optimal representative of the discretization family is highly non-trivial while for large gradients the Scharfetter--Gummel scheme stands out compared to the others.

D. Hömberg, R. Lasarzik, Weak entropy solutions to a model in induction hardening, existence and weak-strong uniqueness, Preprint no. 2671, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2671 .Abstract, PDF (338 kByte)In this paper, we investigate a model describing induction hardening of steel. The related system consists of an energy balance, an ODE for the different phases of steel, and Maxwell's equations in a potential formulation. The existence of weak entropy solutions is shown by a suitable regularization and discretization technique. Moreover, we prove the weak-strong uniqueness of these solutions, i.e., that a weak entropy solutions coincides with a classical solution emanating form the same initial data as long as the classical one exists. The weak entropy solution concept has advantages in comparison to the previously introduced weak solutions, e.g., it allows to include free energy functions with low regularity properties corresponding to phase transitions.

A.F.M. TERElst, A. Linke, J. Rehberg, On the numerical range of sectorial forms, Preprint no. 2667, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2667 .Abstract, PDF (245 kByte)We provide a sharp and optimal generic bound for the angle of the sectorial form associated to a non-symmetric second-order elliptic differential operator with various boundary conditions. Consequently this gives an, in general, sharper H∞-angle for the H∞-calculus on Lp for all p ∈ (1, ∞) if the coefficients are real valued.

R. Lasarzik, Maximal dissipative solutions for incompressible fluid dynamics, Preprint no. 2666, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2666 .Abstract, PDF (170 kByte)We introduce the new concept of maximal dissipative solutions for the Navier--Stokes and Euler equations and show that these solutions exist and the solution set is closed and convex. The concept of maximal dissipative solutions coincides with the concept of weak solutions as long as the weak solutions inherits enough regularity to be unique. A maximal dissipative solution is defined as the minimizer of a convex functional and we argue that this definition bears several advantages.

D. Bothe, P.-É. Druet, Mass transport in multicomponent compressible fluids: Local and global well-posedness in classes of strong solutions for general class-one models, Preprint no. 2658, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2658 .Abstract, PDF (509 kByte)We consider a system of partial differential equations describing mass transport in a multicomponent isothermal compressible fluid. The diffusion fluxes obey the Fick-Onsager or Maxwell- Stefan closure approach. Mechanical forces result into one single convective mixture velocity, the barycentric one, which obeys the Navier-Stokes equations. The thermodynamic pressure is defined by the Gibbs-Duhem equation. Chemical potentials and pressure are derived from a thermodynamic potential, the Helmholtz free energy, with a bulk density allowed to be a general convex function of the mass densities of the constituents. The resulting PDEs are of mixed parabolic-hyperbolic type. We prove two theoretical results concerning the well-posedness of the model in classes of strong solutions: 1. The solution always exists and is unique for short-times and 2. If the initial data are sufficiently near to an equilibrium solution, the well-posedness is valid on arbitrary large, but finite time intervals. Both results rely on a contraction principle valid for systems of mixed type that behave like the compressible Navier- Stokes equations. The linearised parabolic part of the operator possesses the self map property with respect to some closed ball in the state space, while being contractive in a lower order norm only. In this paper, we implement these ideas by means of precise a priori estimates in spaces of exact regularity.

H. Stephan, Millions of Perrin pseudoprimes including a few giants, Preprint no. 2657, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2657 .Abstract, PDF (244 kByte)The calculation of many and large Perrin pseudoprimes is a challenge. This is mainly due to their rarity. Perrin pseudoprimes are one of the rarest known pseudoprimes. In order to calculate many such large numbers, one needs not only a fast algorithm but also an idea how most of them are structured to minimize the amount of numbers one have to test. We present a quick algorithm for testing Perrin pseudoprimes and develop some ideas on how Perrin pseudoprimes might be structured. This leads to some conjectures that still need to be proved. We think that we have found well over 90% of all 20-digit Perrin pseudoprimes. Overall, we have been able to calculate over 9 million Perrin pseudoprimes with our method, including some very large ones. The largest number found has 1436 digits. This seems to be a breakthrough, compared to the previously known just over 100,000 Perrin pseudoprimes, of which the largest have 20 digits. In addition, we propose two sequences that do not provide any pseudoprimes up to 1,000,000,000 at all.

A. Münch, B. Wagner, Self-consistent field theory for a polymer brush. Part II: The effective chemical potential, Preprint no. 2649, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2649 .Abstract, PDF (318 kByte)The most successful mean-field model to describe the collective behaviour of the large class of macromolecular polymers is the self-consistent field theory (SCFT). Still, even for the simple system of a grafted dry polymer brush, the mean-field equations have to be solved numerically. As one of very few alternatives that offer some analytical tractability the strong-stretching theory (SST) has led to explicit expressions for the effective chemical potential and consequently the free energy to promote an understanding of the underlying physics. Yet, a direct derivation of these analytical results from the SCFT model is still outstanding. In this study we present a systematic asymptotic theory based on matched asymtptotic expansions to obtain the effective chemical potential from the SCFT model for a dry polymer brush for large but finite stretching.

A. Münch, B. Wagner, Self-consistent field theory for a polymer brush. Part I: Asymptotic analysis in the strong-stretching limit, Preprint no. 2648, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2648 .Abstract, PDF (854 kByte)In this study we consider the self-consistent field theory for a dry, in- compressible polymer brush, densely grafted on a substrate, describing the average segment density of a polymer in terms of an effective chemical potential for the interaction between the segments of the polymer chain. We present a systematic singular perturbation analysis of the self-consistent field theory in the strong-stretching limit, when the length scale of the ratio of the radius of gyration of the polymer chain to the extension of the brush from the substrate vanishes. Our analysis yields, for the first time, an approximation for the average segment density that is correct to leading order in the outer scaling and resolves the boundary layer singularity at the end of the polymer brush in the strong-stretching limit. We also show that in this limit our analytical results agree increasingly well with our numerical solutions to the full model equations comprising the self-consistent field theory.

P.-É. Druet, A. Jüngel, Analysis of cross-diffusion systems for fluid mixtures driven by a pressure gradient, Preprint no. 2646, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2646 .Abstract, PDF (271 kByte)The convective transport in a multicomponent isothermal compressible fluid subject to the mass continuity equations is considered. The velocity is proportional to the negative pressure gradient, according to Darcy?s law, and the pressure is defined by a state equation imposed by the volume extension of the mixture. These model assumptions lead to a parabolic-hyperbolic system for the mass densities. The global-in-time existence of classical and weak solutions is proved in a bounded domain with no-penetration boundary conditions. The idea is to decompose the system into a porous-medium-type equation for the volume extension and transport equations for the modified number fractions. The existence proof is based on parabolic regularity theory, the theory of renormalized solutions, and an approximation of the velocity field.

A. Glitzky, M. Liero, G. Nika, Analysis of a hybrid model for the electrothermal behavior of semiconductor heterostructures, Preprint no. 2636, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2636 .Abstract, PDF (355 kByte)We prove existence of a weak solution for a hybrid model for the electro-thermal behavior of semiconductor heterostructures. This hybrid model combines an electro-thermal model based on drift-diffusion with thermistor type models in different subregions of the semiconductor heterostructure. The proof uses a regularization method and Schauder's fixed point theorem. For boundary data compatible with thermodynamic equilibrium we verify, additionally, uniqueness. Moreover, we derive bounds and higher integrability properties for the electrostatic potential and the quasi Fermi potentials as well as the temperature.

P. Colli, G. Gilardi, J. Sprekels, Asymptotic analysis of a tumor growth model with fractional operators, Preprint no. 2625, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2625 .Abstract, PDF (341 kByte)In this paper, we study a system of three evolutionary operator equations involving fractional powers of selfadjoint, monotone, unbounded, linear operators having compact resolvents. This system constitutes a generalized and relaxed version of a phase field system of Cahn--Hilliard type modelling tumor growth that has originally been proposed in Hawkins-Daarud et al. (Int. J. Numer. Math. Biomed. Eng. 28(2012), 3--24). The original phase field system and certain relaxed versions thereof have been studied in recent papers co-authored by the present authors and E. Rocca. The model consists of a Cahn--Hilliard equation for the tumor cell fraction φ, coupled to a reaction-diffusion equation for a function S representing the nutrient-rich extracellular water volume fraction. Effects due to fluid motion are neglected. Motivated by the possibility that the diffusional regimes governing the evolution of the different constituents of the model may be of different (e.g., fractional) type, the present authors studied in a recent note a generalization of the systems investigated in the abovementioned works. Under rather general assumptions, well-posedness and regularity results have been shown. In particular, by writing the equation governing the evolution of the chemical potential in the form of a general variational inequality, also singular or nonsmooth contributions of logarithmic or of double obstacle type to the energy density could be admitted. In this note, we perform an asymptotic analysis of the governing system as two (small) relaxation parameters approach zero separately and simultaneously. Corresponding well-posedness and regularity results are established for the respective cases; in particular, we give a detailed discussion which assumptions on the admissible nonlinearities have to be postulated in each of the occurring cases.

P.-É. Druet, Global-in-time existence for liquid mixtures subject to a generalised incompressibility constraint, Preprint no. 2622, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2622 .Abstract, PDF (510 kByte)We consider a system of partial differential equations describing diffusive and convective mass transport in a fluid mixture of N > 1 chemical species. A weighted sum of the partial mass densities of the chemical species is assumed to be constant, which expresses the incompressibility of the fluid, while accounting for different reference sizes of the involved molecules. This condition is different from the usual assumption of a constant total mass density, and it leads in particular to a non-solenoidal velocity field in the Navier-Stokes equations. In turn, the pressure gradient occurs in the diffusion fluxes, so that the PDE-system of mass transport equations and momentum balance is fully coupled. Another striking feature of such incompressible mixtures is the algebraic formula connecting the pressure and the densities, which can be exploited to prove a pressure bound in L1. In this paper, we consider incompressible initial states with bounded energy and show the global existence of weak solutions with defect measure.

C. Grässle, M. Hintermüller, M. Hinze, T. Keil, Simulation and control of a nonsmooth Cahn--Hilliard Navier--Stokes system with variable fluid densities, Preprint no. 2617, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2617 .Abstract, PDF (11 MByte)We are concerned with the simulation and control of a two phase flow model governed by a coupled Cahn--Hilliard Navier--Stokes system involving a nonsmooth energy potential.We establish the existence of optimal solutions and present two distinct approaches to derive suitable stationarity conditions for the bilevel problem, namely C- and strong stationarity. Moreover, we demonstrate the numerical realization of these concepts at the hands of two adaptive solution algorithms relying on a specifically developed goal-oriented error estimator.In addition, we present a model order reduction approach using proper orthogonal decomposition (POD-MOR) in order to replace high-fidelity models by low order surrogates. In particular, we combine POD with space-adapted snapshots and address the challenges which are the consideration of snapshots with different spatial resolutions and the conservation of a solenoidal property.

P. Colli, A. Signori, J. Sprekels, Optimal control of a phase field system modelling tumor growth with chemotaxis and singular potentials, Preprint no. 2614, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2614 .Abstract, PDF (326 kByte)A distributed optimal control problem for an extended model of phase field type for tumor growth is addressed. In this model, the chemotaxis effects are also taken into account. The control is realized by two control variables that design the dispensation of some drugs to the patient. The cost functional is of tracking type, whereas the potential setting has been kept quite general in order to allow regular and singular potentials to be considered. In this direction, some relaxation terms have been introduced in the system. We show the well-posedness of the state system, the Fréchet differentiability of the control-to-state operator in a suitable functional analytic framework, and, lastly, we characterize the first-order necessary conditions of optimality in terms of a variational inequality involving the adjoint variables.

TH. Frenzel, M. Liero, Effective diffusion in thin structures via generalized gradient systems and EDP-convergence, Preprint no. 2601, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2601 .Abstract, PDF (359 kByte)The notion of Energy-Dissipation-Principle convergence (EDP-convergence) is used to derive effective evolution equations for gradient systems describing diffusion in a structure consisting of several thin layers in the limit of vanishing layer thickness. The thicknesses of the sublayers tend to zero with different rates and the diffusion coefficients scale suitably. The Fokker--Planck equation can be formulated as gradient-flow equation with respect to the logarithmic relative entropy of the system and a quadratic Wasserstein-type gradient structure. The EDP-convergence of the gradient system is shown by proving suitable asymptotic lower limits of the entropy and the total dissipation functional. The crucial point is that the limiting evolution is again described by a gradient system, however, now the dissipation potential is not longer quadratic but is given in terms of the hyperbolic cosine. The latter describes jump processes across the thin layers and is related to the Marcelin--de Donder kinetics.

A. Glitzky, M. Liero, G. Nika, An existence result for a class of electrothermal drift-diffusion models with Gauss--Fermi statistics for organic semiconductors, Preprint no. 2593, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2593 .Abstract, PDF (387 kByte)This work is concerned with the analysis of a drift-diffusion model for the electrothermal behavior of organic semiconductor devices. A "generalized Van Roosbroeck” system coupled to the heat equation is employed, where the former consists of continuity equations for electrons and holes and a Poisson equation for the electrostatic potential, and the latter features source terms containing Joule heat contributions and recombination heat. Special features of organic semiconductors like Gauss--Fermi statistics and mobilities functions depending on the electric field strength are taken into account. We prove the existence of solutions for the stationary problem by an iteration scheme and Schauder's fixed point theorem. The underlying solution concept is related to weak solutions of the Van Roosbroeck system and entropy solutions of the heat equation. Additionally, for data compatible with thermodynamic equilibrium, the uniqueness of the solution is verified. It was recently shown that self-heating significantly influences the electronic properties of organic semiconductor devices. Therefore, modeling the coupled electric and thermal responses of organic semiconductors is essential for predicting the effects of temperature on the overall behavior of the device. This work puts the electrothermal drift-diffusion model for organic semiconductors on a sound analytical basis.

A.F.M. TERElst, R. Haller-Dintelmann, J. Rehberg, P. Tolksdorf, On the $L^p$-theory for second-order elliptic operators in divergence form with complex coefficients, Preprint no. 2590, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2590 .Abstract, PDF (383 kByte)Given a complex, elliptic coefficient function we investigate for which values of p the corresponding second-order divergence form operator, complemented with Dirichlet, Neumann or mixed boundary conditions, generates a strongly continuous semigroup on Lp(Ω). Additional properties like analyticity of the semigroup, H∞-calculus and maximal regularity arealso discussed. Finally we prove a perturbation result for real coefficients that gives the whole range of p's for small imaginary parts of the coefficients. Our results are based on the recent notion of p-ellipticity, reverse Hölder inequalities and Gaussian estimates for the real coefficients.

P. Colli, G. Gilardi, J. Sprekels, Longtime behavior for a generalized Cahn--Hilliard system with fractional operators, Preprint no. 2588, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2588 .Abstract, PDF (248 kByte)In this contribution, we deal with the longtime behavior of the solutions to the fractional variant of the Cahn--Hilliard system, with possibly singular potentials, which we recently investigated in the paper "Well-posedness and regularity for a generalized fractional CahnHilliard system". More precisely, we give a complete characterization of the Omega-limit of the phase parameter. The characterization depends on the first eigenvalue of one of the involved operators: if this eigenvalue is positive, then the chemical potential vanishes at infinity, and every element of the Omega-limit is a stationary solution to the phase equation; if it is zero instead, then every element of the Omega-limit solves a problem containing a real function which is related to the chemical potential. Such a function is nonunique and time dependent, in general, as we show by means of an example; however, we give sufficient conditions for it to be uniquely determined and constant.

A. Mielke, T. Roubíček, Thermoviscoelasticity in Kelvin--Voigt rheology at large strains, Preprint no. 2584, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2584 .Abstract, PDF (472 kByte)The frame-indifferent thermodynamically-consistent model of thermoviscoelasticity at large strain is formulated in the reference configuration with using the concept of the second-grade nonsimple materials. We focus on physically correct viscous stresses that are frame indifferent under time-dependent rotations. Also elastic stresses are frame indifferent under rotations and respect positivity of the determinant of the deformation gradient. The heat transfer is governed by the Fourier law in the actual deformed configuration, which leads to a nontrivial description when pulled back into the reference configuration. Existence of weak solutions in the quasistatic setting, i.e. inertial forces are ignored, is shown by time discretization.

R. Rossi, U. Stefanelli, M. Thomas, Rate-independent evolution of sets, Preprint no. 2578, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2578 .Abstract, PDF (475 kByte)The goal of this work is to analyze a model for the rate-independent evolution of sets with finite perimeter. The evolution of the admissible sets is driven by that of a given time-dependent set, which has to include the admissible sets and hence is to be understood as an external loading. The process is driven by the competition between perimeter minimization and minimization of volume changes. In the mathematical modeling of this process, we distinguish the adhesive case, in which the constraint that the (complement of) the `external load' contains the evolving sets is penalized by a term contributing to the driving energy functional, from the brittle case, enforcing this constraint. The existence of Energetic solutions for the adhesive system is proved by passing to the limit in the associated time-incremental minimization scheme. In the brittle case, this time-discretization procedure gives rise to evolving sets satisfying the stability condition, but it remains an open problem to additionally deduce energy-dissipation balance in the time-continuous limit. This can be obtained under some suitable quantification of data. The properties of the brittle evolution law are illustrated by numerical examples in two space dimensions.

A.F.M. TERElst, H. Meinlschmidt, J. Rehberg, Essential boundedness for solutions of the Neumann problem on general domains, Preprint no. 2574, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2574 .Abstract, PDF (220 kByte)Let the domain under consideration be bounded. Under the suppositions of very weak Sobolev embeddings we prove that the solutions of the Neumann problem for an elliptic, second order divergence operator are essentially bounded, if the right hand sides are taken from the dual of a Sobolev space which is adapted to the above embedding.

A. Mielke, J. Naumann, On the existence of global-in-time weak solutions and scaling laws for Kolmogorov's two-equation model of turbulence, Preprint no. 2545, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2545 .Abstract, PDF (467 kByte)This paper is concerned with Kolmogorov's two-equation model for free turbulence in space dimension 3, involving the mean velocity u, the pressure p, an average frequency omega, and a mean turbulent kinetic energy k. We first discuss scaling laws for a slightly more general two-equation models to highlight the special role of the model devised by Kolmogorov in 1942. The main part of the paper consists in proving the existence of weak solutions of Kolmogorov's two-equation model under space-periodic boundary conditions in cubes with positive side length l. To this end, we provide new a priori estimates and invoke existence result for pseudo-monotone operators.

M. Thomas, Coupling of rate-independent and rate-dependent systems with application to delamination processes in solids, Seminar ``Applied and Computational Analysis'', University of Cambridge, UK, October 10, 2019.

N. Rotundo, On a thermodynamically consistent coupling of quantum system and device equations, The 20th European Conference on Mathematics for Industry (ECMI 2018), Minisymposium ``Mathematical Modeling of Charge Transport in Graphene and Low Dimensional Structures'', August 18 - June 22, 2018, Budapest, Hungary, June 19, 2018.

M. Thomas, Analysis and simulation for a phase-field fracture model at finite strains based on modified invariants, Workshop ``Special Materials and Complex Systems'' (SMACS 2018), June 18 - 22, 2018, University of Milan/University of Pavia, Gargnano, Italy, June 18, 2018.

M. Thomas, Analysis and simulation for a phase-field fracture model at finite strains based on modified invariants, Analysis Seminar, University of Brescia, Department of Mathematics, Italy, May 10, 2018.

J. Sprekels, A nonstandard viscous Cahn--Hilliard system with dynamic boundary condition and the DCH, Analysis of Boundary Value Problems for PDEs -- Workshop on the Occasion of the 70th Birthday of Gianni Gilardi, Pavia, Italy, February 20, 2017.

J. Sprekels, Optimal control of PDEs: From basic principles to hard applications, International School ``Frontiers in Partial Differential Equations and Solvers'', May 22 - 25, 2017, University of Pavia, Department of Mathematics, Italy.

J. Sprekels, Well-posedness and optimal control of a nonstandard Cahn--Hilliard system with dynamic boundary condition, Fudan University, School of Mathematical Sciences, China, April 10, 2017.

N. Ahmed, A numerical study of residual based variational multiscale methods for turbulent incompressible flow problems, American University of the Middle East, Dasman, Kuwait, November 2, 2017.

M. Heida, Large deviation principle for a stochastic Allen--Cahn equation, 9th European Conference on Elliptic and Parabolic Problems, May 23 - 27, 2016, University of Zurich, Institute of Mathematics, Gaeta, Italy, May 25, 2016.

M. Thomas, Rate-independent evolution of sets, INdAM-ISIMM Workshop on Trends on Applications of Mathematics to Mechanics, September 5 - 8, 2016, The International Society for the Interaction of Mechanics and Mathematics (ISIMM), Rome, Italy, September 6, 2016.

S.P. Frigeri, On a diffuse interface model of tumor growth, 9th European Conference on Elliptic and Parabolic Problems, May 23 - 27, 2016, University of Zurich, Institute of Mathematics, Gaeta, Italy, May 23, 2016.

M. Hintermüller, Recent trends in optimal control problems with nonsmooth structures, Computational Methods for Control of Infinite-dimensional Systems, March 14 - 18, 2016, Institute for Mathematics and its Applications, Minneapolis, USA, March 14, 2016.

A. Mielke, On the geometry of reaction and diffusion, INdAM-ISIMM Workshop on Trends on Applications of Mathematics to Mechanics, September 5 - 8, 2016, The International Society for the Interaction of Mechanics and Mathematics (ISIMM), Rome, Italy, September 7, 2016.

D. Peschka, Droplets on liquids and their long way into equilibrium, Minisymposium ``Recent Progress in Modeling and Simulation of Multiphase Thin-film Type Problems'' of the 8th International Congress on Industrial and Applied Mathematics (ICIAM 2015), August 10 - 14, 2015, International Council for Industrial and Applied Mathematics, Beijing, China, August 12, 2015.

M.H. FarshbafShaker, A deep quench approach to the optimal control of an Allen--Cahn equation with dynamic boundary conditions, National Institute for Mathematical Sciences, Division of Computational Mathematics, Daejeon, Korea (Republic of), May 20, 2015.

M.H. FarshbafShaker, Multi-material phase field approach to structural topology optimization and its relation to sharp interface approach, University of Tokyo, Graduate School of Mathematical Sciences, Japan, October 6, 2015.

A. Mielke, A mathematical approach to finite-strain viscoplasticity, Intensive Period on Variational Methods for Plasticity and Dislocations, March 16 - 20, 2015, International School of Advanced Studies (SISSA), Trieste, Italy, March 20, 2015.

A. Mielke, Abstract approach to energetic solutions for rate-independent solutions, Intensive Period on Variational Methods for Plasticity and Dislocations, March 16 - 20, 2015, International School of Advanced Studies (SISSA), Trieste, Italy, March 18, 2015.

A. Mielke, The multiplicative strain decomposition in finite-strain elastoplasticity, Intensive Period on Variational Methods for Plasticity and Dislocations, March 16 - 20, 2015, International School of Advanced Studies (SISSA), Trieste, Italy, March 17, 2015.

K. Disser, Asymptotic behaviour of a rigid body with a cavity filled by a viscous liquid, Second Workshop of the GAMM Activity Group on "Analysis of Partial Differential Equations", September 29 - October 1, 2014, Universität Stuttgart, Lehrstuhl für Analysis und Modellierung, October 1, 2014.

K. Disser, Asymptotic behaviour of a rigid body with a cavity filled by a viscous liquid, Autumn School and Workshop on Mathematical Fluid Dynamics, October 27 - 30, 2014, Universität Darmstadt, International Research Training Group 1529, Bad Boll, October 28, 2014.

C. Kreisbeck, Thin-film limits of functionals on A-free vector fields and applications, XIX International Symposium on Trends in Applications of Mathematics to Mechanics (STAMM 2014), September 8 - 11, 2014, The International Society for the Interaction of Mechanics and Mathematics (ISIMM), Poitiers, France, September 9, 2014.

E. Rocca, ``Entropic'' solutions to a thermodynamically consistent PDE system for phase transitions and damage, Symposium on Trends in Application of Mathematics to Mechanics (STAMM), September 8 - 11, 2014, International Society for the Interaction of Mechanics and Mathematics (ISIMM), Poitiers, France, September 9, 2014.

E. Valdinoci, Concentration solutions for a nonlocal Schroedinger equation, Kinetics, Non Standard Diffusion and the Mathematics of Networks: Emerging Challenges in the Sciences, May 7 - 16, 2014, The University of Texas at Austin, Department of Mathematics, USA, May 14, 2014.

A. Mielke, On the microscopic origin of generalized gradient structures for reaction-diffusion systems, XIX International Symposium on Trends in Applications of Mathematics to Mechanics (STAMM 2014), September 8 - 11, 2014, The International Society for the Interaction of Mechanics and Mathematics (ISIMM), Poitiers, France, September 11, 2014.

P. Gussmann, Linearized elasticity as $Gamma$-limit of finite elasticity in the case of cracks, 84th Annual Meeting of the International Association of Applied Mathematics and Mechanics, Section ``Applied Analysis'', March 18 - 22, 2013, University of Novi Sad, Serbia, March 20, 2013.

S. Heinz, On a way to control oscillations for a special evolution equation, Conference ``Nonlinearities'', June 10 - 14, 2013, University of Warsaw, Institute of Mathematics, Male Ciche, Poland, June 11, 2013.

M. Thomas, Damage and delamination processes in thermo-viscoelastic materials, 84th Annual Meeting of the International Association of Applied Mathematics and Mechanics, Young Reserchers' Minisymposium ``Analytical and Engineering Aspects in the Material Modeling of Solids'', March 18 - 22, 2013, University of Novi Sad, Serbia, March 19, 2013.

M. Thomas, A stress-driven local solution approach to quasistatic brittle delamination, Seminar on Functional Analysis and Applications, International School of Advanced Studies (SISSA), Trieste, Italy, November 12, 2013.

M. Thomas, A model for rate-independent, brittle delamination in thermo-visco-elasticity, International Workshop on Evolution Problems in Damage, Plasticity, and Fracture: Mathematical Models and Numerical Analysis, September 19 - 21, 2012, University of Udine, Department of Mathematics, Italy, September 21, 2012.

A. Petrov, On a 3D model for shape-memory alloys, Workshop ``Rate-independent Systems: Modeling, Analysis, and Computations'', August 30 - September 3, 2010, Banff International Research Station for Mathematical Innovation and Discovery (BIRS), Canada, September 2, 2010.

CH. Kraus, An inhomogeneous, anisotropic and elastically modified Gibbs-Thomson law as singular limit of a diffuse interface model, 81st Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM), March 22 - 26, 2010, Karlsruhe, March 23, 2010.

J. Sprekels, Technical and mathematical problems in the Czochralski growth of single crystals, Workshop ``New Directions in Simulation, Control and Analysis for Interfaces and Free Boundaries'', January 31 - February 6, 2010, Mathematisches Forschungsinstitut Oberwolfach, February 1, 2010.

S. Heinz, A model for the evolution of laminates, 80th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2009), Young Researchers Minisymposium ``Mathematics and Mechanics of Microstructure Evolution in Finite Plasticity'', February 9 - 13, 2009, Gdansk University of Technology, Poland, February 10, 2009.

A. Petrov, On the existence and error bounds for space-time discretizations of a 3D model for shape-memory alloys, Lisbon University, Center for Mathematics and Fundamental Applications, Portugal, September 17, 2009.

A. Glitzky, Discrete Sobolev--Poincaré inequalities using the $W^1,p$ seminorm in the setting of Voronoi finite volume approximations, International Conference on Elliptic and Parabolic Equations, November 30 - December 4, 2009, WIAS, December 3, 2009.

H.-Chr. Kaiser, A thermodynamic approach to transient Kohn--Sham theory, 100th Statistical Mechanics Conference, December 13 - 18, 2008, Rutgers, The State University of New Jersey, New Brunswick, USA, December 16, 2008.

H.-Chr. Kaiser, On drift-diffusion Kohn--Sham theory, 79th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2008), session ``Applied Analysis'', March 31 - April 4, 2008, University of Bremen, April 1, 2008.

H.-Chr. Kaiser, A drift-diffusion model of transient Kohn--Sham theory, First Joint International Meeting between the American Mathematical Society and the Polish Mathematical Society, Special Session ``Mathematics of Large Quantum Systems'', July 31 - August 3, 2007, University of Warsaw, Poland, August 3, 2007.

A. Petrov, Mathematical result on the stability of elastic-plastic systems with hardening, European Conference on Smart Systems, October 26 - 28, 2006, Researching Training Network "New Materials, Adaptive Systems and their Nonlinearities: Modelling, Control and Numerical Simulation" within the European Commission's 5th Framework Programme, Rome, Italy, October 27, 2006.

Preprints im Fremdverlag

A. Mielke, M.A. Peletier, D.R.M. Renger, A generalization of Onsager's reciprocity relations to gradient flows with nonlinear mobility, Preprint no. arXiv:1510.06219, Cornell University Library, arXiv.org, 2015.AbstractOnsager's 1931 `reciprocity relations' result connects microscopic time-reversibility with a symmetry property of corresponding macroscopic evolution equations. Among the many consequences is a variational characterization of the macroscopic evolution equation as a gradient-flow, steepest-ascent, or maximal-entropy-production equation. Onsager's original theorem is limited to close-to-equilibrium situations, with a Gaussian invariant measure and a linear macroscopic evolution. In this paper we generalize this result beyond these limitations, and show how the microscopic time-reversibility leads to natural generalized symmetry conditions, which take the form of generalized gradient flows.

D.A. Gomes, S. Patrizi, Obstacle mean-field game problem, Preprint no. arXiv:1410.6942, Cornell University Library, arXiv.org, 2014.AbstractIn this paper, we introduce and study a first-order mean-field game obstacle problem. We examine the case of local dependence on the measure under assumptions that include both the logarithmic case and power-like nonlinearities. Since the obstacle operator is not differentiable, the equations for first-order mean field game problems have to be discussed carefully. Hence, we begin by considering a penalized problem. We prove this problem admits a unique solution satisfying uniform bounds. These bounds serve to pass to the limit in the penalized problem and to characterize the limiting equations. Finally, we prove uniqueness of solutions.

S. Neukamm, A. Gloria, F. Otto, An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations, Preprint no. 41, Max-Planck-Institut für Mathematik in den Naturwissenschaften, 2013.AbstractWe establish an optimal, linear rate of convergence for the stochastic homogenization of discrete linear elliptic equations. We consider the model problem of independent and identically distributed coefficients on a discretized unit torus. We show that the difference between the solution to the random problem on the discretized torus and the first two terms of the two-scale asymptotic expansion has the same scaling as in the periodic case. In particular the L2-norm in probability of the H1-norm in space of this error scales like ε, where ε is the discretization parameter of the unit torus. The proof makes extensive use of previous results by the authors, and of recent annealed estimates on the Greens function by Marahrens and the third author.